A geometry problem by Saikat Sengupta

Geometry Level 4

Let $ABC$ be a triangle and $D$ be the midpoint of $BC$ . Suppose the angle bisector of $\angle ADC$ is the tangent to the circumcircle of triangle $ABD$ at $D$ , find the measure of angle $A$ in degrees.

90 30 60 120 100 0 1

2 solutions

Vishwash Kumar Γξω
Jan 20, 2017

Saikat Sengupta
Jan 14, 2017

Let P be the circumcentre of circumcircle Γ of 4ABC. Let the tangent at D to Γ intersect AC in E. Then P D ⊥ DE. Since DE bisects ∠ADC, this implies that DP bisects ∠ADB. Hence the circumcenter and the incenter of 4ABD lies on the same line DP. This implies that DA = DB. Thus DA = DB = DC and hence D is the circumcenter of 4ABC. This gives ∠A = 90◦ .

Asked in RMO 2016 1st question WB Region. Isnt it? I gave it this year.

Md Zuhair - 4 years, 4 months ago

Na Karnataka regions.

Vishwash Kumar ΓΞΩ - 4 years, 4 months ago

Also in WB Region bro..

Md Zuhair - 4 years, 4 months ago

A geometry problem by Saikat Sengupta

2 solutions

0 pending reports