Question

in an acute triangle pqr angle pqr is equal to 45 degree then prove that PR square is equal to PQ square + Q R square - 4 into area of triangle pqr

Answer 1

construct PN perpendicular to QR

in triangle PNQ, angle Q=45

so QN=PN

PQ^2+QR^2-4*areaPQR=PN^2+QN^2+(QN+NR)2-4*0.5*QR*PN

=2PN^2+PN^2+NR^2+2*PN*NR-2*QR*PN

=2PN^2+PR^2+2PN.NR-2(PN+NR)*PN

=2PN^2-2PN^2+2PN*NR-2PN*NR+PR^2