If P is the Orthocentre of Triangle ABC. Show that A,B,C are the orthocenter of Triangles BPA, PCA, PBC?

Answer 1

AP extended is perpendicular to BC.
BP extended is perpendicular to AC.
CP extended is perpendicular to AB.

then
CB is perpendicular to PA.
CA is perpendicular to PB.
making C the orthocenter of ∆PAB

AC is perpendicular to PB.
AB is perpendicular to PC.
making A the orthocenter of ∆PBC

BA is perpendicular to PC.
BC is perpendicular to PA.
making B the orthocenter of ∆PAC

Answer 2

P divides the triangle ABC into three different right angled triangles BPA,PCA,PBC. As we know that the orthocentre of the right angled triangle is the right angled vertex. If we draw perpendiculars from vertices, they intersect at right angled vertex.
Thus, P also is the orthocenter of PBA, PBC, PCA.