# Understand the problem

Let be a triangle and be cevians concurrent at a point . Suppose each of the quadrilaterals and has both circumcircle and incircle. Prove that is equilateral and coincides with the center of the triangle.

##### Source of the problem

Indian team selection test 2018

##### Topic

Geometry

##### Difficulty Level

Hard

##### Suggested Book

Challenge and Thrill of Pre-college Mathematics by B J Venkatachala, C R Pranesachar, K N Ranganathan and V Krishnamurthy

# Start with hints

Do you really need a hint? Try it first!

Show that is cyclic. Afterwards, study the consequences of this result.

As , hint 1 means that . Hence is the orthocentre of . Let be the circumcenter of and be its incentre. From the information given in the second link of hint 1, we see that are colinear.Also, as , lies on .

As lies on , is the internal bisector of . As is also an altitude, this means that . By symmetry, and . Hence the triangle is equilateral.

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#### Math Olympiad Program

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