Inside an equilateral triangle labelled ABC a point D is placed. The distance from D to the 3 vertices is 5, 12 and 13. What is the area of ABC?
1) Use the regular polygon tool to create an equilateral triangle with vertices A(0,0),B(16,0). Make sure you have auxiliary objects on.
2) Draw a circle with radius 5 using A as the center, color the circumference red. Do the same for B and C with radii of 12 and 13. Color them blue and green.
3) Pull B until the 3 circles appear to intersect at one point. Then use the shift arrows to make fine adjustments. Refer to the first diagram.
4) Now use the intersection tool and find the point of intersection of the red and green circles and the blue and green circles. If you have been careful then those two points will be indistinguishable to the eye.
5) The point of intersection will have two names, E and F. Hide F and draw line segments AE, BE,CE.
6) Look at g,h and i. I have 5,13 and 12.006059. Pretty good, how did you do? Now is the time to adjust B to get as close or closer than I did.
7) Check the Area of poly1, I have 118.233329, the actual answer is,
We are quite close!
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.