Apollonius Point Ap is a special point in a triangle. It is defined as the intersection point of the lines connecting the vertices of the triangle with the points of tangency of the extra-inscribed circles with the circumscribed around them. Connected with the task of Apollonius . The Encyclopedia of Triangle Centers is referred to as the center of a triangle under the name X (181).
Content
An example of applying the Apollonius point to the solution of the Apollonius problem
The task of Apollonius is to construct a circle with the help of a compass and a ruler that touches these three circles. One of the variants of this problem, when the third circle touches the three internal ones externally, is solved by introducing the Apollonius point Ap [1] [2] .
- The Apollonius Ap point in the Encyclopedia of Triangle Centers is referred to as the center of the triangle under the name X (181).
- In the framework of this problem, the circle of Apollonius (not to be confused with the circles of Apollonius ) is the circle that touches the three extra-inscribed circles outside the triangle in an internal way (see the green circle in the figure).
Apollonius Circle
Apollonius Circle Definition
- Let triangle ABC be given. Let the inscribed circles of triangle ABC , opposite the vertices A , B, and C , be E A , E B , E C , respectively (see the figure). Then the Apollonius circle E (shown in green on the right) touches inwardly three extra-inscribed circles of triangle ABC at points E A , E B, and E C , respectively (see figure) [3] .
- The solution of the above particular Apollonius problem is the indicated circle E tangent to the three given circles E A , E B and E C externally.
Apollonius Circle Radius
The radius of the circle of Apollonius is , where r is the radius of the inscribed circle and s is the semiperimeter of the triangle. [four]
Apollonius Ap Point Definition
- The Apollonius point Ap or X (181) is defined as follows:
Let A ' , B' and C ' be the points of tangency of the circle of Apollonius E with the corresponding extra-inscribed circles. Then the lines AA ' , BB' and CC ' intersect at one point Ap , which is called the Apollonius point of triangle ABC .
Note
In the figure, the indicated Apollonius point Ap is depicted as the intersection point of three perpendiculars to the sides of the triangle ABC , omitted from the tangent points A ' , B' and C ' with the corresponding extra-inscribed circles of the triangle ABC formed by the joint pairwise tangent lines of the three circles E A , E mentioned above B and E C. Although this point Ap lies at the intersection of the three segments AA ' , BB' and CC ' , but they are not perpendicular to the sides of the triangle. Indeed, its projections on the sides of the triangle ABC are the vertices of an equilateral triangle, and the perpendiculars to the sides of the triangle intersect in its orthocenter. The projections of the orthocenter on the sides of the triangle are not vertices of an equilateral triangle. The orthocenter and Apollonius point Ap coincide only in an equilateral triangle. Other triangles do not match.
Trilinear coordinates
The trilinear coordinates of the Apollonius Ap point :
- ( a ( b + c ) 2 / ( b + c - a ): b ( c + a ) 2 / ( c + a - b ): c ( a + b ) 2 / ( a + b - c )
- = ((sin A cos ( B / 2 - C / 2)) 2 : (sin B cos ( C / 2 - A / 2)) 2 : (sin C cos ( A / 2 - B / 2)) 2 )
Notes
- β Kimberling, Clark Apollonius Point . Date of appeal May 16, 2012.
- β C. Kimberling; Shiko Iwata; Hidetosi Fukagawa. Problem 1091 and Solution (Eng.) // Crux Mathematicorum : journal. - 1987. - Vol. 13 . - P. 217-218 .
- β Darij Grinberg, Paul Yiu. The Apollonius Circle as a Tucker Circle // Forum Geometricorum. - 2002. - Vol. 2 . - S. 175-182 .
- β Milorad R. StevanoviΒ΄c. The Apollonius circle and related triangle centers // Forum Geometricorum. - 2003. - Issue. 3 . - S. 187-195. .