# Online Geometry Problem 86. Intouch and Extouch Triangles, Equal Area, Contact Triangles. Level: High School, College, Mathematics Education

 In the figure below, given a triangle ABC, Si is the area of the contact or intouch triangle DEF, Se is the area of the extouch triangle GHM. Prove that: .     "A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such expert experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime." George Polya, 1944    HINTS: 1. The contact triangle of a triangle ABC, also called the intouch triangle or Gergonne triangle, is the triangle DEF formed by the points of tangency of the incircle of triangle ABC with triangle ABC. 2. The extouch triangle of a triangle ABC is the triangle GHM formed by the points of tangency of the triangle ABC with its excircles.   3. TANGENT TO A CIRCLE Proposition. Two tangent segments to a circle from an external point are congruent.   4. Semiperimeter s, Side and Incircle Formula   5. Semiperimeter s, Side and Excircle Formula   6. AREA OF A TRIANGLE: Proposition: The area of a triangle equals one-half the product of the length of a side and the length of the altitude to that side. Side Angle Side Formula: The SAS formula = ½ (side1 × side2) × sine(included angle). 6. See Problems 82, 85.
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