# 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 A1C1 is tangent to the incircle at point T, and we have AA2 = CC1 and CC2 = AA1. The problem is to prove that B2B3 = BB1.

## 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.