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Belarus 2022 geometry
Problem
The incircle of the right triangle is tangent to the hypotenuse at point and is tangent to the legs and at points and respectively. Points and are symmetric to with respect to the lines and . Find the angle where is the incenter of the triangle . (Mikhail Karpuk)
Solution
Denote . Let's do some angle-chasing. Therefore , i.e. is the center of the circumcircle of the triangle . Hence
Similarly, . Then
$$ \angle C_1IC_2 = 90^\circ + \angle QIC_1 + \angle RIC_2 = 180^\circ - \frac{\angle A + \angle B}{2} = 135^\circ.
Similarly, . Then
$$ \angle C_1IC_2 = 90^\circ + \angle QIC_1 + \angle RIC_2 = 180^\circ - \frac{\angle A + \angle B}{2} = 135^\circ.
Final answer
135°
Techniques
Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleTangentsAngle chasing