Some Triangle Centers Associated with the Tritangent Circles

Nikolaos Dergiades, Juan Salazar
2009 Forum Geometricorum   unpublished
We investigate two interesting special cases of the classical Apollo-nius problem, and then apply these to the tritangent circles of a triangle to find pair of perspective (or homothetic) triangles. Some new triangle centers are constructed. 1. An interesting construction We begin with a simple construction of a special case of the classical Apollonius problem. Given two circles O(r), O ′ (r ′) and an external tangent L, to construct a circle O 1 (r 1) tangent to the circles and the line, with
more » ... and the line, with point of tangency X between A and A ′ , those of (O), (O ′) and L (see Figure 1). A simple calculation shows that AX = 2 √ r 1 r and XA ′ = 2 √ r 1 r ′ , so that AX : XA ′ = √ r : √ r ′. The radius of the circle is r 1 = 1 4 AA ′ √ r + √ r ′ 2. O O ′ A A ′ X O1 L Figure 1 From this we design the following construction. Construction 1. On the line L, choose two points P and Q be points on opposite sides of A such that P A = r and AQ = r ′. Construct the circle with diameter P Q to intersect the line OA at F such that O and F are on opposite sides of L. The intersection of O ′ F with L is the point X satisfying AX : XA ′ = √ r : √ r ′. Let M be the midpoint of AX. The perpendiculars to OM at M , and to L at X intersect at the center O 1 of the required circle (see Figure 2).