# Geometry Problem 1503: A Triangle, Its Incircle, a Tangent, Congruence, and a Perpendicular

In the
given figure, point D is the incenter of
triangle ABC. Line A_{1}C_{1} is tangent to the incircle at
point T, and we have AA_{2} =
CC_{1} and CC_{2} = AA_{1}. The problem is to
prove that B_{2}B_{3} = BB_{1}.

## Definitions

- A triangle is a three-sided polygon that is one of the most basic shapes in geometry.
- A tangent to a circle is a straight line that
touches the circle at exactly one point. This point of contact is called the
point of tangency. A tangent is perpendicular to the radius drawn to the
point of tangency.
- Incircle is a circle inscribed within another shape touching every side of it at one point.
- Incenter is the center point of an incircle.
- Perpendicular lines are two lines that intersect at a right angle (90 degrees).
- Congruent figures have the same size and shape, and they can be matched up exactly.
- A triangle transversal line is a line that intersects two or more sides of a triangle. It is called a "transversal" because it cuts across or "transverses" the sides of the triangle.