1954 AMC 12 Problem 19

Question

If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle:

are always equal to

60^\circ

are always one obtuse angle and two unequal acute angles

are always one obtuse angle and two equal acute angles

are always acute angles

are always unequal to each other

Join the incenter to two adjacent points of tangency

The angle of the contact triangle opposite vertex angle

A

is

90^\circ-\frac{A}{2}

The radii to two adjacent tangency points are perpendicular to the corresponding sides. The central angle between those radii is

180^\circ-A.

The relevant angle of the contact triangle is half the intercepted arc, giving

90^\circ-\frac A2.

The other two angles are

90^\circ-\frac{B}{2}

and

90^\circ-\frac{C}{2}.

Since every triangle angle lies strictly between

0^\circ

and

180^\circ,

all three are acute.

Thus, the correct answer is D.

Problem 19 in other years