Consider a line that simultaneously bisects the area and the perimeter of a given triangle.
Only two such lines exists for a triangle of side lengths 7,8,9.
Find the acute angle θ between the two lines.
In triangle ABC, let AB=7, AC=8, and BC=9.
Draw a circle of radius 6 on C, crossing AC at D and BC at E. DE is then the first line.
Draw two circles of radius (63sqrt(2)) and (6+3sqrt(2)) on B, crossing AB at F and BC at G respectively. FG is the second line.
The acute angle between these lines is of 85.6082 degrees.
Note: The controlling equation is (t(P/4))^2((P/2)^22*ab)/4, where P is the perimeter, and ab are the sides adjoining the angle on which the circles are to be drawn. The solutions to this quadratic are the radii of the circles. The remaining, hypothetical, solution would involve circles of radius (63/sqrt(2), 6+3/sqrt(2)) on A, but the larger circle does not intersect a side of ABC. Hence the only solutions are those given above.
Edited on September 12, 2020, 8:05 am

Posted by broll
on 20200912 07:52:22 