DE

E denoting the projection on the chord BC of contact points and {B',C'} denoting correspondingly the projections on the tangents {AB,AC}.

Follows immediately from the similarity of triangles DBC, DB'E, DEC'.

Notice that quadrangles DEBB' and DC'CE are similar. If triangle DEC' is modified so that it remains similar to itself while C' glides on AC then E will move on BC (see Similarly_Rotating.html ). Analogous statement holds for triangle DB'E.

DA'*DB'*DC' = DA''*DB''*DC''.

Here {A',B',C'} are the projections of D on the sides of ABC and {A'',B'',C''} the projections of D on the tangents at {A,B,C} correspondingly.

It suffices to apply the basic relation of (1) for each ange of the tangential triangle: DB'

DX

[1] Project P on two consecutive sides, {AB,AD} say, to create triangle PGH. Analogously project on the other two consecutive sides {CB,CD} to create triangle PEF. The two triangles are similar. This follows easily by angle chasing argument.

[2] The circumcenters {O

[3] Analogous statements hold for the triangles created similarly by selecting the other pair of opposite vertices {B,D}.

[4] The circumcenters of the four circumcircles {O

[5] Project D on pairs of opposite sides {AB,CD} to create triangle PJI and {AD,BC} to create PKL.

area(PLK)/area(PJI) = sin(x)/sin(y),

where {x,y} are the angles formed by opposite sides. Thus this area-quotient is independent of the position of P on the circle.

[2,3,4] Follows from the fact that {PA,PB,PC,PD} are diameters of the circumcircles of triangles considered.

[5] Is a consequence of the previous similarities.

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