https://www.cut-the-knot.org/proofs/ptolemy.shtml#Mahavira

It's the same as Pierre's remark but perhas a little more detailed. What I want you to do is to make a diagram where you mark the angle theta and think of what cos(theta) signifies, OK?

Sorry, still nothing! I do not understand anything! Are you sure there is not a flaw in the argument? I have a feeling there is. Could you please send me a diagram and an explanation? You may draw it by hand and fax it through at +30-210-7219225 if this is easier for you.

The formula is proven very easily by using the cosine rule.

m^2=b^2+c^2-2.b.c.cosC
=a^2+d^2-2.a.d.cosA=a^2+d^2-2.a.d.cos(180-C)=a^2+d^2+2.a.d.cosC
Multiply the first with ad and the second with bc, add them up and you will arrive at m^2=(ab+cd).(ac+bd)/(ad+bc)

I am really surprised. I do not understand what cosè signifies. There might be an error in the proof.

Have a bash at the probability question I posed. I think I have got an answer, it depends on what one is really asking. I look forward to see the replies.

ab = hR - can you verify that? It'says that in a triangle the product of two sides equals the product of the diameter circumscribed circle and the altitude to the third side.

1. In a quadrilateral, you have two pairs of triangles sharing a base (the third side in 1). Bases are the diagonals of the quadrilateral. Consider one pair at a time, say, ab = ht and cd = gt, where h is the altitude to from the ab vertex to the diagonal m, I belive. g is the altitude from the cd vertex to the sam diagonal.

2. Each of these triangles contains a piece of the diagonal n which plays the hypotenuse, one in a triangle with side h, the other in a triangle with side g. The triangles are similar. You can write

   h = cos(theta)×(one piece) and
   g = cos(theta)×(the other piece).

   On summing up you get

   h + g = cos(theta)·n

   Thus adding ab = ht and cd = gt gives you

   nt·cos(theta) = ab + cd.

Sorry if I confused m and n.