Quadrilateral WXYZ has right angles at angle W and angle Y and an acute angle at angle X. Altitudes are dropped from X and Z to diagonal WY, meeting WY at O

<OXW  = < XWO  = 90                             <ZYP  = < OYX  = 90

<PWZ = <XWO  =90                                <ZYP  = <YZP  =90

So < OXW = < PWZ                                 So < OYX  = < YZP

And <XOW  = < ZPW  = 90                      And <XOY = <ZPY

So  ΔXOW  ~ Δ WPZ                                So  ΔXOY  ~  ΔPYZ

XO / OW  = WP / PZ                                XO /OY = PY /PZ

XO * PZ  = WP * OW                                 XO* PZ  = OY* PY

So

WP * OW   = OY* PY

OW / PY  = OY/ WP

OW / PY = [ OP + PY] / [OP + OW]

OP*OW + OW^2  = OP*PY + PY^2

OP[OW − PY] +  OW^2  −  PY^2  = 0

OP[OW − PY]  +  [OW − PY] [[OW + PY]  = 0

[OW − PY] [ OP + (OW + PY) ]  = 0          

And.......since OP , PY and OW  are  all > 0  .....the second factor can never  = 0

So  [OW − PY]  must = 0   →    OW  = PY