Question

Please solve q 99?

Answer 1

`DP+EP+PF=5sqrt3` regardless where P is in the triangle.
This is called Viviani's theorem: https://en.wikipedia.org/wiki/Viviani%27s_theorem

Explanation:

ABC is the triangle. It's height `h=5sqrt3` (use Pythagoras on for instance ADC)

Remember that the area of a triangle is `(base*height)/2`.

Now all three triangles APC, BPC and APB have a base = 10 since all sides are equal and 10. Their total area is equal to the area of ABC

therefore `(side*CM)/2=(side*h)/2`
=`(side*DP)/2+(side*EP)/2*(side*FP)/2`

Therefore `DP+EP+PF=MC=5sqrt3` regardless where P is in the triangle.

Answer 2

Another one from Rahul's Book!

I'd say that given we're given choices, the result doesn't depend on our choice of `P`.

Let's pick `P=G,` the centroid/orthocenter. We get an altitude `5^2+h^2=10^2` or `h=5 sqrt{3}`. The medians meet `2/3` from each vertex, and the remaining piece of `1/3h` is perpendicular to the sides. We have three of these to add up, they add to `h`, choice (2).