Point of Minimal Distance -Equilateral Triangle

Using GEOMETRY ONLY, For an equilateral triangular region, for which points is the sum of the distances to the sides of the triangle minimal? Please show me and not point to a web site. Thanks much!

Proof 1/r = 1/a +1/b + 1/c Incircle

Let r be the radius of the incircle of triangle ABC, and a, b, c be the radii of the excircles opposite vertices A,B, and C respectively. Write a proof illustrating the fact that 1/r = 1/a +1/b + 1/c. Please do not just direct me to a web page. I would like a detailed breakdown step by step on how this is done. Thanks much!

Centroid of Triangular Region- Balance Point

Using GEOMETRY only, I need to prove that the centroid of a triangular region (assuming uniform density) is the balance point. Trick is that I can't use the "equal areas" argument. I need a step-by-step explanation so I can walk through the process. Please don't send me to a website as I can do that for free. Thanks much!!

Area of a quadrilateral

What is the area of a quadrilateral with sides of length 16, 12, 15, and 25, and a right angle between the sides with lengths 16 and 12?

Pre calc

Need equation for: Ellipse Center at(0,4); focus at(8,4); vertex at (6,7)