ive tried 127, 4 and 19.72 and they were wrong... i need help :(

Question

In triangle , point  is on side  such that    and the circumcircles of triangles  and  have the same radius. Find the area of triangle .

Answer 1

OK, I got you an approximate answer using analytic geometry and DESMOS.

I do NOT see any simple geometric or trigonometric way to do this.

If you play around with graph paper and DESMOS, starting with the points (-6.5,0)and (6.5,0), the slope of the side through (6.5,0) with the 2 parts 14 and 4 is approximately -2.5.

Once you get that circumcenters are approximately (5,0) and (-5,0) and the other 2 points of the triangle are (1.301,12.999) and (7.986,-3.714).

I did not compute the area but it is a straight forward but messy computation to get it, using either Heron's formula or the determinant form for the area given the co-ordinates.

Good luck.

P.S.

If when you review this in class you get a simpler way to solve this problem, please let me know...tutors can still learn too!

5/15/2020

I got it now. Missed it until now.

Angle ACP and angle ABC are equal (measured by the same arc).

Angle APB and angle PAB are also equal for the same reason.

Now you have the lengths of all the sides and you get the area by Heron's formula!

Sorry it took so long.