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.
Triangle, Incircle, Tangent, Congruence, Perpendicular


  • 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.

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