A geometry problem by abhishek anand

Geometry Level 5

. Let ABCD be a convex quadrilateral with 6 DAB = 6 BDC = 90β—¦ . Let the incircles of triangles ABD and BCD touch BD at P and Q, respectively, with P lying in between B and Q. If AD = 999 and P Q = 200 then what is the sum of the radii of the incircles of triangles ABD and BDC?


The answer is 799.

Abhishek Anand by abhishek anand , 20, India

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2 solutions

Ayush G Rai
Oct 8, 2016

Assume that the incircle of 𝐴 𝐡 𝐷 𝐴𝐡𝐷 touches 𝐴 𝐷 𝐴𝐷 at 𝐸 𝐸 and the inradius of 𝐴 𝐡 𝐷 𝐴𝐡𝐷 and 𝐡 𝐢 𝐷 𝐡𝐢𝐷 be π‘Ÿ 1 π‘Ÿ1 and π‘Ÿ 2. π‘Ÿ2. Thus 𝐴 𝐸 = π‘Ÿ 1 β‡’ 𝐸 𝐷 = 999 βˆ’ π‘Ÿ 1 β‡’ 𝐷 𝑃 = 𝐷 𝐸 = 999 βˆ’ π‘Ÿ 1 β‡’ π‘Ÿ 2 = 𝐷 𝑄 = 𝐷 𝑃 βˆ’ 𝑃 𝑄 = 799 βˆ’ π‘Ÿ 1 β‡’ π‘Ÿ 1 + π‘Ÿ 2 = 799 . 𝐴𝐸 = π‘Ÿ1 β‡’ 𝐸𝐷 = 999 - π‘Ÿ1 β‡’ 𝐷𝑃 = 𝐷𝐸 = 999 - π‘Ÿ1 β‡’ π‘Ÿ2 = 𝐷𝑄 =𝐷𝑃 - 𝑃𝑄 = 799 - π‘Ÿ1 β‡’ π‘Ÿ1 + π‘Ÿ2 = \boxed{799}.

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Hrishik Mukherjee
Jun 16, 2015

abhishek anand Put a solution please

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