Looks can be deceiving

Geometry Level 4

Construct a triangle $ABC$ with $\angle BAC=74^{\circ}$ . Consider the mixtilinear circle with respect to $A$ (The circle tangent to $AB$ , $AC$ and the circumcircle internally). Let the points of tangency to $AB$ and $AC$ be $X$ and $Y$ . Let the midpoint of $XY$ be $M$ . Find $\angle MAB$ in degrees.

If you think no set angle occurs, then answer 1337.

The answer is 37.

1 solution

Sharky Kesa
Apr 25, 2016

Call the centre of this mixtilinear circle $\Gamma_a$ . Because $AB$ and $AC$ are tangent to it, $\Gamma_a$ must lie on the angle bisector of $\angle BAC$ . Furthermore, we must have $XY \perp A\Gamma_a$ .

Note that a line passing through the centre of a circle is perpendicular to a chord (of the same circle) iff the line passes through the midpoint of the chord. Thus, we must have that $M$ lies on the angle bisector of $\angle BAC$ . Thus, $\angle MAB = 37^{\circ}$ .

Investigation : There's something quite neat about $M$ . It is actually the incentre of triangle $ABC$ . See if you can prove it. My hint is to use Pascal's Theorem.

Looks can be deceiving

The answer is 37.

1 solution

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