Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to those sides is equivalent to the triangle formed by the points of contact of the sides of the traingle with the inscribed circle
Is it from pre college mathematics
To prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to those sides is equivalent to the triangle formed by the points of contact of the sides of the triangle with the inscribed circle, we need to show that the two triangles have equal side lengths.
Let's consider a triangle ABC, and let D, E, and F be the points of contact of the excircles corresponding to sides BC, CA, and AB, respectively. Let G, H, and I be the points of contact of the inscribed circle with sides BC, CA, and AB, respectively.
To prove the equivalence, we will show that DG = GI, EI = IF, and FH = HD.
1. DG = GI:
By the definition of the excircle, segment DG is the distance from point D to the point of contact of the incircle with side BC. Similarly, segment GI is the distance from point G to the point of contact of the incircle with side BC. Since both segments are radii of the same circle (the incircle), they have the same length. Therefore, DG = GI.
2. EI = IF:
By the same reasoning as above, segment EI is the distance from point E to the point of contact of the incircle with side CA, and segment IF is the distance from point F to the point of contact of the incircle with side CA. Again, both segments are radii of the incircle, so they have the same length. Hence, EI = IF.
3. FH = HD:
Similarly, segment FH is the distance from point F to the point of contact of the incircle with side AB, and segment HD is the distance from point H to the point of contact of the incircle with side AB. Again, both segments are radii of the incircle, so they have the same length. Therefore, FH = HD.
Since we've shown that the corresponding sides of the two triangles have equal lengths, we can conclude that the triangle formed by the points of contact of the sides with the excircles is equivalent to the triangle formed by the points of contact of the sides with the inscribed circle.
To prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to those sides is equivalent to the triangle formed by the points of contact of the sides of the triangle with the inscribed circle, we can follow these steps:
1. Let's start by considering a triangle ABC with its corresponding excircle centers Ia, Ib, and Ic. The excircle center Ia is the center of the excircle tangent to side BC, and similarly, Ib and Ic are the centers of the excircles tangent to sides AC and AB, respectively.
2. Draw the tangent lines from the vertices A, B, and C to the corresponding excircles. Let the intersection points of these tangent lines with the sides of the triangle be A', B', and C', respectively.
3. Now, let's consider the incircle of triangle ABC with the center I and the points of contact with the sides be D, E, and F.
4. It is a well-known property that the points of contact of the sides of a triangle with its incircle lie on the angle bisectors of the triangle. Therefore, AD, BE, and CF are the angle bisectors of triangle ABC.
5. Similarly, it can be proven that the points of contact A', B', and C' lie on the angle bisectors of triangle ABC with respect to the excircles.
6. Now, based on the property of angle bisectors, we can conclude that triangle A'B'C' is similar to triangle DEF because A'D, B'E, and C'F are corresponding angle bisectors.
7. Additionally, since triangle ABC is similar to triangle DEF (all the angle bisectors are the same), we can conclude that triangle A'B'C' is also similar to triangle ABC.
8. Since two triangles are similar if and only if their corresponding sides are proportional in length, we can conclude that the sides of triangle ABC are proportional in length to the sides of triangle A'B'C'.
Hence, we have proven that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to those sides is indeed equivalent (similar) to the triangle formed by the points of contact of the sides of the triangle with the inscribed circle.