Geometry Problem 1457: Altitudes, Circles, Similarity, Product of the Inradii Lengths

Let AHA, BHB, CHC be the altitudes of a triangle ABC. The extensions of AHA, BHB, CHC intersect the circumcircle O at A1, B1, C1. Let r1, r2, r3, r4, r5, r6 represent the length of the inradii of the triangles AB1H, CB1H, CA1H, BA1H, BC1H, AC1H. Prove that \(r_1\cdot r_3\cdot r_5=r_2\cdot r_4\cdot r_6\).

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Static Diagram of problem 1457

Dynamic Geometry 1457: Altitudes, Circles, Similarity, Product of the Inradii Lengths. Using GeoGebra

Poster of the problem 1457 using iPad Apps

Poster of Problem 1457, Altitudes, Circles, Similarity, Product of the Inradii Lengths, GeoGebra, iPad


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Geometry Problem 1457 Solution(s)