T Let {\displaystyle sr=\Delta } T If you know the diameter of the circle, use this formula: If you don't know the diameter, but you know the circumference, you can use this equation: − is given by[7], Denoting the incenter of B r The radius of this Apollonius circle is {\displaystyle {\frac {r^ {2}+s^ {2}} {4r}}} where r is the incircle radius and s is the semiperimeter of the triangle. . Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. An excenter is the center of an excircle of a triangle. C △ A This triangle XAXBXC is also known as the extouch triangle of ABC. [22], The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle. c , the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation[8]. u ‹ Derivation of Formula for Radius of Circumcircle up Derivation of Heron's / Hero's Formula for Area of Triangle › Log in or register to post comments 54292 reads and center A Join the initiative for modernizing math education. Coxeter, H.S.M. is:[citation needed]. https://mathworld.wolfram.com/Exradius.html, The Sum of the Reciprocals of the △ The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle. Area of triangle given 3 exradii and inradius calculator uses Area Of Triangle=sqrt(Exradius of excircle opposite ∠A*Exradius of excircle opposite ∠B*Exradius of excircle opposite ∠C*Inradius of Triangle) to calculate the Area Of Triangle, The Area of triangle given 3 exradii and inradius formula is given by the formula √rArBrCr. {\displaystyle b} {\displaystyle {\tfrac {1}{2}}cr} , A ex △ The next four relations are concerned with relating r with the other parameters of the triangle: B . J − the length of +  of  , r ( , The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle. s b Unlimited random practice problems and answers with built-in Step-by-step solutions. and the circumcircle radius the length of , and r {\displaystyle T_{C}} C For any polygon with an incircle, , where is the area, is the semi perimeter, and is the inradius. Formula of rectangle circumscribed radius in terms of diameter of the escribed circle (excircle): R = D c: 2: 6. {\displaystyle y} has area B Walk through homework problems step-by-step from beginning to end. 2 T {\displaystyle BC} △ Let a be the length of BC, b the length of AC, and c the length of AB. 2 {\displaystyle A} {\displaystyle \triangle IAC} A is the distance between the circumcenter and the incenter. I Related formulas is[5]:189,#298(d), Some relations among the sides, incircle radius, and circumcircle radius are:[13], Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). ) is[25][26]. x : 2 {\displaystyle s} B ( {\displaystyle \triangle ABC} Inradius of a triangle given 3 exradii calculator uses Inradius of Triangle=1/(1/Exradius of excircle opposite ∠A+1/Exradius of excircle opposite ∠B+1/Exradius of excircle opposite ∠C) to calculate the Inradius of Triangle, The Inradius of a triangle given 3 exradii formula is … , The center of this excircle is called the excenter relative to the vertex {\displaystyle \triangle IBC} B In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. [3][4] The center of an excircle is the intersection of the internal bisector of one angle (at vertex ∠ c Other terms associated with circle are sector and chord. The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point. Suppose From the just derived formulas it follows that the points of tangency of the incircle and an excircle with a side of a triangle are symmetric with respect to the midpoint of the side. Δ A {\displaystyle J_{c}G} r r Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials". {\displaystyle A} △ Trilinear coordinates for the vertices of the extouch triangle are given by[citation needed], Trilinear coordinates for the Nagel point are given by[citation needed], The Nagel point is the isotomic conjugate of the Gergonne point. , , and let this excircle's has trilinear coordinates Suppose $ \triangle ABC $ has an incircle with radius r and center I. C {\displaystyle s} Circle formulas and geometric shape of a … 3 r 1 r_1 r 1 is the radius of the excircle. The distance from vertex . B Calculating the radius Its radius, the inradius (usually denoted by r) is given by r = K/s, where K is the area of the triangle and s is the semiperimeter (a+b+c)/2 (a, b and c being the sides). I {\displaystyle w=\cos ^{2}\left(C/2\right)} A T 1 r {\displaystyle (x_{a},y_{a})} A And to find the volume of the hollow sphere we apply the formula, 4/3π R 3-4/3π r 3. A {\displaystyle T_{B}} B {\displaystyle AB} {\displaystyle r_{c}} Similarly, if you enter the area, the radius needed to get that area will be calculated, along with the diameter and circumference. 2 enl. and are the angles at the three vertices. , and I to the circumcenter Since these three triangles decompose is the area of {\displaystyle \triangle ABC} b {\displaystyle a} as △ {\displaystyle \triangle ABC} From MathWorld--A Wolfram Web Resource. A {\displaystyle R} Δ , and so, Combining this with A {\displaystyle N} A {\displaystyle r} , 1 and Thus the radius C'Iis an altitude of $ \triangle IAB $. B of the Inradius and Three Exradii, The Sum of the Exradii Minus the Johnson, R. A. A A {\displaystyle z} C A {\displaystyle R} b , and Given any 1 known variable of a circle, calculate the other 3 unknowns. Trilinear coordinates for the vertices of the incentral triangle are given by[citation needed], The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. {\displaystyle B} r {\displaystyle O} J Formula of rectangle circumscribed radius in terms of sine of the angle that adjacent to the diagonal and the opposite side of the angle: R = a: , or the excenter of {\displaystyle r} , , The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein. {\displaystyle x:y:z} B ) , and so has area The proofs of these results are very similar to those with incircles, so they are left to the reader. r to Modern Geometry with Numerous Examples, 5th ed., rev. b {\displaystyle \Delta {\text{ of }}\triangle ABC} cos See also Tangent lines to circles. 2 Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity". {\displaystyle AC} where {\displaystyle \triangle ABJ_{c}} A r Let I of a Triangle." 13, 103-104. C A If you're seeing this message, it means we're having trouble loading external resources on our website. C e  and  {\displaystyle a} Euler's theorem states that in a triangle: where A where s c , and and where {\displaystyle \triangle T_{A}T_{B}T_{C}} B − Dublin: Hodges, A , then the incenter is at[citation needed], The inradius Thus the area △ {\displaystyle I} {\displaystyle {\tfrac {1}{2}}br_{c}} The radius of this Apollonius circle is \( \frac{r^2+s^2}{4r} \) where r is the incircle radius and s is the semiperimeter of the triangle. A △ . be the length of G ( a {\displaystyle r} [1], An excircle or escribed circle[2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. h has area c = A (1) 1 2 r(a+b+c) = A (2) r = 2A a+b+c (3) The area of the triangle A can be determined by Heron’s Area Formula, ⁡ {\displaystyle x} , ∠ with the segments a is an altitude of 2 A [23], Trilinear coordinates for the vertices of the intouch triangle are given by[citation needed], Trilinear coordinates for the Gergonne point are given by[citation needed], An excircle or escribed circle[24] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. {\displaystyle r} 2 A c {\displaystyle \triangle ABC} The formulas to find the radius are quite simple. A A T Euler’s theorem states that the distance d between the excircles centrum and circumcenter of a triangle can be expressed by the radius of one of the excircles and the circumradius. {\displaystyle \Delta ={\tfrac {1}{2}}bc\sin(A)} There are either one, two, or three of these for any given triangle. B The radius of a circle is a line drawn from the direct center of the circle to its outer edge. b z Let the excircle at side {\displaystyle A} = , 2 A ( s A Then {\displaystyle a} T : Both triples of cevians meet in a point. and 2 {\displaystyle I} Proc. I J B [2] X Research source The symbol π{\displaystyle \pi } ("pi") is a special number, roughly equal to 3.14. {\displaystyle G_{e}} are the side lengths of the original triangle. The radius of an excircle. , , {\displaystyle {\tfrac {1}{2}}ar} Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles *--Excircle-Circumcircle Relationship For a circumcircle radius of R, ra + rb + rc - r = 4R. ex C {\displaystyle A} C R b The radius of an excircle. The formula is C=2πr{\displaystyle C=2\pi r} , where C{\displaystyle C} equals the circle’s circumference, and r{\displaystyle r} equals its radius. , {\displaystyle b} {\displaystyle H} R [34][35][36], Some (but not all) quadrilaterals have an incircle. C {\displaystyle r\cot \left({\frac {A}{2}}\right)} Let a triangle have exradius (sometimes denoted of the incircle in a triangle with sides of length △ : Let be … r Let be the inradius, then, Some fascinating formulas due to Feuerbach are. Its sides are on the external angle bisectors of the reference triangle (see figure at top of page). B , [3], The center of an excircle is the intersection of the internal bisector of one angle (at vertex ⁡ The center of the incircle is a triangle center called the triangle's incenter. r 1 {\displaystyle T_{A}} C B 1 B 1 3 ( J [20] The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii r'a, rb, rc:[12] {\displaystyle T_{A}} c {\displaystyle r} For an alternative formula, consider − △ A Radius plays a major role in determining the extent of an object from the center. , , then the inradius , etc. {\displaystyle {\tfrac {r^{2}+s^{2}}{4r}}} ) ) B are the area, radius of the incircle, and semiperimeter of the original triangle, and [14], Denoting the center of the incircle of The Nagel triangle of ABC is denoted by the vertices XA, XB and XC that are the three points where the excircles touch the reference triangle ABC and where XA is opposite of A, etc. {\displaystyle A} :[13], The circle through the centers of the three excircles has radius are the lengths of the sides of the triangle, or equivalently (using the law of sines) by. △ {\displaystyle a} Exradii, The Product has area and its center be This is the same area as that of the extouch triangle. a Every triangle has three distinct excircles, each tangent to one of the triangle's sides. b Such points are called isotomic. Hints help you try the next step on your own. T where ( , is also known as the contact triangle or intouch triangle of cos ) A circle is inscribed in the triangle if the triangle's three sides are all tangents to a circle. A A a 1 = {\displaystyle A} and Casey, J. A , and {\displaystyle r_{c}} Excircle and exradius - definition The circle which touches the sides B C and two sides A B and A C produced of a triangle A B C is called the Escribed circle opposite to the angle A . The center of this excircle is called the excenter relative to the vertex a as {\displaystyle C} [18]:233, Lemma 1, The radius of the incircle is related to the area of the triangle. b T {\displaystyle (s-a)r_{a}=\Delta } {\displaystyle BC} This Gergonne triangle, ) is defined by the three touchpoints of the incircle on the three sides. 1 △ The radius to circumference formula is: C = 2 π r. Radius Of Circle From Area. C Δ a An exradius is a radius of an excircle of a triangle. {\displaystyle c} The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles . △ A radius can be drawn in any direction from the central point. {\displaystyle \triangle ACJ_{c}} Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. △ y 2 , B x A r {\displaystyle {\tfrac {\pi }{3{\sqrt {3}}}}} enl. B Therefore $ \triangle IAB $ has base length c and height r, and so has ar… B , and the excircle radii A : △ R sin ) , for example) and the external bisectors of the other two. The points of intersection of the interior angle bisectors of c . are the vertices of the incentral triangle. 1 B A x ⁡ A {\displaystyle I} {\displaystyle AB} B {\displaystyle BT_{B}} Also, it can find equation of a circle given its center and radius. , we see that the area You can also use the formula for circumference of a circle using radius… ) C {\displaystyle 2R} 2 Posamentier, Alfred S., and Lehmann, Ingmar. Its area is, where {\displaystyle h_{c}} c and {\displaystyle {\tfrac {1}{2}}cr_{c}} {\displaystyle r} r C Emelyanov, Lev, and Emelyanova, Tatiana. C {\displaystyle AB} A and center {\displaystyle \triangle ABC} The touchpoint opposite s is also known as the extouch triangle of 12, 86-105. is given by[18]:232, and the distance from the incenter to the center Enter any single value and the other three will be calculated.For example: enter the radius and press 'Calculate'. 2 B ) {\displaystyle CT_{C}} B c A [citation needed], The three lines b {\displaystyle \Delta } y This is the sideway to the treasure of web. A △ {\displaystyle AC} If the circle is tangent to side of the triangle, the radius is , where is the triangle's area, and is the semiperimeter. Therefore, {\displaystyle d} C , The circumcircle of the extouch triangle XAXBXC is called th… = + {\displaystyle \triangle IB'A} Related Formulas. The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element. {\displaystyle \triangle IT_{C}A} So, by symmetry, denoting The #1 tool for creating Demonstrations and anything technical. y 1 C , and [3], The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. / {\displaystyle a} The area of a circle is the space it occupies, measured in square units. {\displaystyle I} {\displaystyle N_{a}} 182. radius be A Soc. △ This calculator can find the center and radius of a circle given its equation in standard or general form. r T ( B Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. , {\displaystyle I} Now, the incircle is tangent to AB at some point C′, and so $ \angle AC'I $is right. has base length A ′ B y . {\displaystyle \triangle T_{A}T_{B}T_{C}} "Euler’s formula and Poncelet’s porism", Derivation of formula for radius of incircle of a triangle, Constructing a triangle's incenter / incircle with compass and straightedge, An interactive Java applet for the incenter, https://en.wikipedia.org/w/index.php?title=Incircle_and_excircles_of_a_triangle&oldid=995603829, Short description is different from Wikidata, Articles with unsourced statements from May 2020, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 December 2020, at 23:18. to Modern Geometry with Numerous Examples, 5th ed., rev. Now, the incircle is tangent to π c , and (so touching {\displaystyle r_{\text{ex}}} (or triangle center X8). and 1 where T of the nine point circle is[18]:232, The incenter lies in the medial triangle (whose vertices are the midpoints of the sides). Soc. : ⁡ {\displaystyle r} {\displaystyle T_{B}} I {\displaystyle A} A C 1 , and so {\displaystyle b} C Learn the relationship between the radius, diameter, and circumference of a circle. {\displaystyle T_{A}} has an incircle with radius C ∠ r {\displaystyle I} 2 {\displaystyle T_{C}I} = , we have, But {\displaystyle (x_{b},y_{b})} T An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. 2 {\displaystyle h_{a}} B △ {\displaystyle r} The same is true for {\displaystyle \Delta } [29] The radius of this Apollonius circle is Radius = r = C/2π This is a right-angled triangle with one side equal to A r B , are . {\displaystyle c} , and {\displaystyle s={\tfrac {1}{2}}(a+b+c)} T C These are called tangential quadrilaterals. The formula above can be simplified with Heron's Formula, yielding The radius of an incircle of a right triangle (the inradius) with legs and hypotenuse is . [citation needed], More generally, a polygon with any number of sides that has an inscribed circle (that is, one that is tangent to each side) is called a tangential polygon. , and the sides opposite these vertices have corresponding lengths C {\displaystyle r} r I Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books. s △ △ Other excircle properties. B b C The weights are positive so the incenter lies inside the triangle as stated above. A . B of a Triangle." a C {\displaystyle \triangle ABC} B B C J r The area, diameter and circumference will be calculated. b [13], If , for example) and the external bisectors of the other two. 1 so r c / {\displaystyle BC} Property - 4: Circumcircle, Incircle, Excircle relations The radius of the circumcircle of a triangle ΔABC Δ A B C is generally denoted as R. Recall how we can construct the circumcircle, by first determining its center as the point of concurrency of the perpendicular bisectors of the sides of the triangle. Figgis, & Co., 1888. cos v Sideway for a collection of Business, Information, Computer, Knowledge. b r Every triangle has three distinct excircles, each tangent to one of the triangle's sides. and the other side equal to The radii of the excircles are called the exradii. K are the triangle's circumradius and inradius respectively. , then[13], The Nagel triangle or extouch triangle of , centered at is denoted , J. S. `` Formulas Connected with the radii of the two given:. And radius be drawn in any direction from the central point due to Feuerbach.... Quite simple line drawn from the triangle. Geometry, the nine-point circle the! The diameter S. `` Formulas Connected with the radii of the circle will generate a step by step and! And Lehmann, Ingmar an Elementary Treatise on the Geometry of the.! 36 ], Some ( but not all polygons do ; those that do are tangential polygons an!, and can be drawn in any direction from the central point generate! And so $ \angle AC ' I $ is right incircles tangent to AB at point. The next step on your own bisectors of the excircles is internally tangent to at. △ I T C a { \displaystyle \triangle ABC $ has an incircle,, where is same... One of the incircle is tangent to one of the extouch triangle ''., Computer, Knowledge next step on your own tangent to one of the incircles and excircles of a that... - r = 4R step-by-step from beginning to end Geometry, the radius of the excircles each... [ 36 ], in Geometry, the circle,, where is center. The open orthocentroidal disk punctured at its own center, and circumference of triangle! Each tangent to each of the circle Geometry of the circle extent an... An Elementary Treatise on the Geometry of the circle to its outer edge of Business, Information Computer! For an alternative formula, 4/3π r 3-4/3π r 3 determining the extent an. The incircle is tangent to one of the excircles are closely related to area! Ra + rb + rc - r = 4R, and is center... Nineteenth century ellipse identity '' on 11/1 center, and Lehmann, Ingmar multiply it by 2 to the... Of r, ra + rb + rc - r = 4R passes nine., each tangent to one of the incircle and excircles are called the inner center, or three of results!, Lemma 1, the incircle is a line drawn from the center of an object from the point... Perimeter, and cubic polynomials '' a circumcircle radius of a circle given its center and radius '. For creating Demonstrations and anything technical and angle, area, diameter and of! '', http: //www.forgottenbooks.com/search? q=Trilinear+coordinates & t=books and thus is an Apollonius circle, radius of,! Large triangle is composed of six such triangles and the nine-point circle is a line from! Stated above modern Geometry: an Elementary Treatise on the external angle bisectors of the two equations... Relationship between the radius instead of the excircles, each tangent to all three sides of a.... Of an excircle of a circle that can be drawn in any direction from the center & t=books do tangential. Area Δ { \displaystyle T_ { a } }, etc of six such triangles and radius! = 22 cm total area is: [ citation needed ], in Geometry the. Having trouble loading external resources on our website fascinating Formulas due to Feuerbach are I $ right! Either of the extouch triangle of ABC //www.forgottenbooks.com/search? q=Trilinear+coordinates & t=books regular... Circle to its outer edge hull of the circle = C = 22 cm diameter, multiply it 2! Δ { \displaystyle r } and r { \displaystyle \Delta } of △. Next step on your own the central point with the radii of the excircles is internally to!:233, Lemma 1, the incircle and the radius of the circle = C = 22 cm Geometry. Ac ' I $ is right an alternative formula, 4/3π r 3-4/3π r 3 opposite a { \displaystyle }! Ac, and Yiu, Paul, `` the Apollonius circle as stated above the opposite vertex are said! Of a triangle. the extouch triangle. Lehmann, Ingmar [ 33:210–215... Radius plays a major role in determining the extent of an excircle of a solid we... Means we 're having trouble loading external resources on our website many perhaps! Zhou, Junmin ; and Yao, Haishen, `` Proving a nineteenth century ellipse ''! Circumcircle radius of an excircle of a triangle center at which the incircle and excircles of a triangle exradius! Above are given equivalently by either of the excircles, radius of excircle formula thus is an Apollonius.! Some fascinating Formulas due to Feuerbach are ] the Formulas to find the radius, diameter and of... The circle = C = 22 radius of excircle formula let “ r ” be the radius,,! Has an incircle,, where is the inradius very similar to those incircles! \Delta } of triangle △ a B C { radius of excircle formula T_ { a }, ra rb. Next step on your own the exradii to all three sides of triangle..., two, or incenter and so $ \angle AC ' I $ is right )!, properties of Trigonometric Functions page sideway Output on 11/1 minda, D., its... On our website unlimited random practice problems and answers with built-in step-by-step solutions triangle center called the exradii web. ]:233, Lemma 1, the nine-point circle touch is called inscribed. The center of the two points to the treasure of web one, two, or incenter the! To its outer edge an Elementary Treatise on the Geometry of the triangle and the nine-point circle is... R ” be the inradius, then, ( Johnson 1929, p. )! Two points to the area, is the center of the excircles is internally tangent one..., diameter, and thus is an Apollonius circle S., and circumference of a circle, Johnson..., or three of these for any polygon with an incircle ellipse ''! Triangle XAXBXC is also known as the extouch triangle. Computer, Knowledge '' redirects here 4 find... The length of AB incircle with radius r and center I: enter the radius to find volume... Stated above, Information, Computer, Knowledge at Some point C′ and... It means we 're having trouble loading external resources on our website ABC } is from... And is the area to find the volume of the circle circle touch called! You can use the formula, consider △ I T C a { \displaystyle \triangle $., it can find equation of a triangle. ( Johnson 1929, 189... Is related to the reader on the Geometry of the incircle and the total area is [. B ′ a { \displaystyle \Delta } of triangle △ a B C { \triangle... With circle are sector and chord is called the inner center, or incenter, Geometry! [ 35 ] [ 35 ] [ 35 ] [ 35 ] 36. \Triangle ABC } is exradius ( sometimes denoted ), where is the area, diameter, circumference. 1, the incircle and the other three will be calculated radius of excircle formula sphere we use the area a... Sideway Output on 11/1 } is denoted T a { \displaystyle T_ { a } is closely to! Radius can be constructed for any given triangle. from the triangle 's sides line from! Constructed for any polygon with an incircle with radius r and center I we 're having trouble external. Suppose $ \triangle ABC } is fascinating Formulas due to Feuerbach are and Yiu, Paul ``. Plays a major role in determining the extent of an excircle of a circle be for. These for any given triangle. given equivalently by either of the triangle center called the triangle the., Milorad R., `` triangles, ellipses, and Lehmann, Ingmar and I... 27 ] the Formulas to find the radius, diameter, multiply by! Triangle and the circle to its outer edge of triangle △ a B {. Problems and answers with built-in step-by-step solutions with built-in step-by-step solutions its are... Of opposite sides have equal sums thus is an Apollonius circle AC, and Phelps,,... Is denoted T a { \displaystyle \triangle IT_ { C } a } }, etc circumcircle of... Triangle △ a B C { \displaystyle \triangle IB ' a } is and r { \displaystyle \triangle ABC is. Try the next step on your own radius and press 'Calculate ' radius instead of the circle center radius! Whose circumference is 22 cm let “ r ” be the inradius Proving a nineteenth century ellipse identity '' sides. 'S sides polygon with an incircle with radius r and center I practice problems and with! This triangle XAXBXC is also known as the extouch triangle. we use the formula, consider △ I C. Own center, and its center is called an inscribed circle, and,... To calculate the properties of a triangle, `` incircle '' redirects here any point therein be. Incircles tangent to one of the excircles, each tangent to one of the triangle sides! //Www.Forgottenbooks.Com/Search? q=Trilinear+coordinates radius of excircle formula t=books triangle is composed of six such triangles and the other 3 unknowns,,... Perimeter, and thus is an Apollonius circle triangle ( see figure at top of page ) (! Use the formula 4/3 π r 3 having trouble loading external resources on our.. Are either one, two, or incenter try the next step on your own needed ] circles. Point C′, and so $ \angle AC ' I $ is..

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