QED.

Bob

]]>I just did 1/2 product of the diagonals. Sketchpad does have an area function, but I've never used it.

Bob

]]>There are three equations ( two from cosine rule and the straight line equation) and there are three unknowns (R,r and alpha). So it should be possible to eliminate two unknowns to find the third. Both the cosine rule equations have the same term in cos alpha but in one it is negative. So adding the equations together (LHS1 + LHS2 = RHS1 + RHS2) gets alpha out straight away. But I eliminated R first before the adding so I had a quadratic in r straight away.

Thanks for giving me the start. I had stared at that diagram for ages without thinking of that. Too distracted trying to use angle properties.

I think the expression for r and R could be left in surd form and then the area obtained without any recourse to decimals. Might try it later. That way you get to show the answer is exactly 180.

Bob

]]>Your idea and then I followed a different path:

Triangle DBA

Triangle ABC

Using R = 12 + r this becomes

Eliminate alpha by adding

Bob

]]>B

]]>Sounds brilliant but too many steps missing for my little brain.

Which triangle(s) are you using for the cosine rule?

Bob

]]>where alpha is the acute angle of the rhombus. We can get from that equation the cosine of the angle and thus also its sine. We then use the formula for calculating the area of parallelogram using its sides and internal angle.]]>