Question

The lengths of the medians through acute angles of a right-angled triangle are 3 and 4. Find the area of the triangle.

Answer 1

AD=3
CE=4
Apollonian theorem
`(AB)^2+(AC)^2=2((AD)^2+(BD)^2)`
`b^2+c^2=2(9+a_2/4)-(1)`
`b^2+a^2=2(c^2/4+16)-(2)`
`b^2=a^2+c^2`
`2b^2+a^2+c^2=2(a^2/4+c^2/4+25)`
`2b^2+b^2=2(b^2/4+25)`
`3b^2=b^2/2+50`
`5/2b^2=50` `b^2=20`
`a=2sqrt(11/3)`
`c=4/sqrt3`
Area of D=`1/2*4/sqrt3*2sqrt(11/3)`
`=4/3sqrt11`.