- Dear friends:

Let X be touch point at BC with the extangent circle to excircles

(I_b) and (I_c) of ABC triangle. Y and Z are similarly the touch

points on AC and AB. If we consider the triangle UVW performed by the

polars of vertex of ABC wrt the opposite excircles. Seems XYZ and UVW

are perspective at P. Is P in ETC?

Thanks in advance

Best regards

Juan Carlos > If we consider the three small extangent circles (O_1),(O_2),(O_3)

and

> NPC of ABC, then the radicals axis to the three circles (O_1),(O_2),

Dear Juan Carlos and all

> (O_3) with NPC performed a triangle QST, hence QST and ABC are

> perspective at P'.

> Is P'in ETC?

Related to this problem I can give a construction that uses inversion

and the coordinates of your point P', not in ETC.

The construction can be found as a solution of problem 301 in Ricardo

Barroso's online magazine

http://www.aloj.us.es/rbarroso/trianguloscabri/sol/sol301garcap/

(server is slow, so you may need try it several times)

I gave there a formula for the radius of (O_1)

A rough translation of the Spanish text follows.

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Let K, L be the tangency points of (Ic),(Ib) with line BC.

We consider an inversion with center K and radius KL (or power KL^2).

The triangle KLIb is rectangle at L, so the circle (Ib) is orthogonal

to the inversion circle and thus invarible by the inversion. The circle

(Ic) pass trough the inversion center, so circle (Ic) inverts in a

line, namely, if KM is a diameter of (Ic), then circle (Ic) inverts in

the parallel to BC through M', the inverse of M.

The circle we want is the inverse of the circle (T) tangent to this

parallel, the line BC and the circle (Ib).

Then the center T must be in the median parallel to two lines. Let this

median parallel intersect KM at N. Then T is KN + rb away from Ib, and

this let us get the center T.

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To end this long post I give the coordinates of P', not in ETC.

Search Number is 1.367790355

We name 2 sqrt((s-b)(s-c)) as RA, and RB, RC the same ciclically.

{(a + b - c)*(a - b + c)*(b + c)*(2*a^4 + a^2*b^2 + b^4 + 2*a^2*b*c -

3*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*a^3*RA - 2*a*b^2*RA +

2*a*c^2*RA)*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 - 2*a^5*c -

3*a^4*b*c + 2*a^3*b^2*c + a^2*b^3*c - 2*a*b^4*c - 2*a^3*b*c^2 -

2*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 + 6*a^2*b*c^3 +

2*a*b^2*c^3 - b^3*c^3 + a*b*c^4 - 3*b^2*c^4 - 2*a*c^5 -

3*b*c^5 - a^5*RB - a^4*b*RB + a^3*b^2*RB + a^2*b^3*RB - a^4*c*RB -

4*a^3*b*c*RB - a^2*b^2*c*RB + 2*a*b^3*c*RB +

2*a^3*c^2*RB + 2*a^2*b*c^2*RB - a*b^2*c^2*RB + b^3*c^2*RB +

2*a^2*c^3*RB + 4*a*b*c^3*RB + b^2*c^3*RB - a*c^4*RB - b*c^4*RB -

c^5*RB)*(2*a^5*b - 4*a^3*b^3 + 2*a*b^5 + 3*a^5*c - a^4*b*c -

6*a^3*b^2*c + 2*a^2*b^3*c + 3*a*b^4*c - b^5*c + 3*a^4*c^2 -

2*a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + a^3*c^3 -

a^2*b*c^3 - a*b^2*c^3 + b^3*c^3 + a^2*c^4 + 2*a*b*c^4 +

b^2*c^4 + a^5*RC + a^4*b*RC - 2*a^3*b^2*RC - 2*a^2*b^3*RC +

a*b^4*RC + b^5*RC + a^4*c*RC - 4*a^3*b*c*RC - 2*a^2*b^2*c*RC +

4*a*b^3*c*RC + b^4*c*RC - a^3*c^2*RC + a^2*b*c^2*RC + a*b^2*c^2*RC

- b^3*c^2*RC - a^2*c^3*RC - 2*a*b*c^3*RC - b^2*c^3*RC),

(a - b - c)*(a + b - c)*(a + c)*(a^4*b^2 + a^3*b^3 + 3*a^2*b^4 +

3*a*b^5 + 2*a^4*b*c - a^3*b^2*c - 2*a^2*b^3*c - a*b^4*c +

2*b^5*c + a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - 6*a*b^3*c^2 +

a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - a^2*c^4 +

3*a*b*c^4 - a*c^5 + 2*b*c^5 - a^3*b^2*RA - a^2*b^3*RA + a*b^4*RA +

b^5*RA - 2*a^3*b*c*RA + a^2*b^2*c*RA - 4*a*b^3*c*RA +

b^4*c*RA - a^3*c^2*RA + a^2*b*c^2*RA - 2*a*b^2*c^2*RA -

2*b^3*c^2*RA - a^2*c^3*RA + 4*a*b*c^3*RA - 2*b^2*c^3*RA + a*c^4*RA +

b*c^4*RA + c^5*RA)*(a^4 - 3*a^2*b^2 + 2*b^4 + 2*a*b^2*c - 2*a^2*c^2

+ b^2*c^2 + c^4 + 2*a^2*b*RB - 2*b^3*RB - 2*b*c^2*RB)*

(2*a^5*b - 4*a^3*b^3 + 2*a*b^5 + 3*a^5*c - a^4*b*c - 6*a^3*b^2*c +

2*a^2*b^3*c + 3*a*b^4*c - b^5*c + 3*a^4*c^2 - 2*a^3*b*c^2 +

2*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + a^3*c^3 - a^2*b*c^3 -

a*b^2*c^3 + b^3*c^3 + a^2*c^4 + 2*a*b*c^4 + b^2*c^4 + a^5*RC +

a^4*b*RC - 2*a^3*b^2*RC - 2*a^2*b^3*RC + a*b^4*RC + b^5*RC +

a^4*c*RC - 4*a^3*b*c*RC - 2*a^2*b^2*c*RC + 4*a*b^3*c*RC +

b^4*c*RC - a^3*c^2*RC + a^2*b*c^2*RC + a*b^2*c^2*RC - b^3*c^2*RC -

a^2*c^3*RC - 2*a*b*c^3*RC - b^2*c^3*RC),

-((a + b)*(a - b - c)*(a - b + c)*(a^4*b^2 + a^3*b^3 + 3*a^2*b^4 +

3*a*b^5 + 2*a^4*b*c - a^3*b^2*c - 2*a^2*b^3*c - a*b^4*c +

2*b^5*c + a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - 6*a*b^3*c^2 +

a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - a^2*c^4 +

3*a*b*c^4 - a*c^5 + 2*b*c^5 - a^3*b^2*RA - a^2*b^3*RA + a*b^4*RA +

b^5*RA - 2*a^3*b*c*RA + a^2*b^2*c*RA - 4*a*b^3*c*RA +

b^4*c*RA - a^3*c^2*RA + a^2*b*c^2*RA - 2*a*b^2*c^2*RA -

2*b^3*c^2*RA - a^2*c^3*RA + 4*a*b*c^3*RA - 2*b^2*c^3*RA + a*c^4*RA +

b*c^4*RA + c^5*RA)*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 - 2*a^5*c

- 3*a^4*b*c + 2*a^3*b^2*c + a^2*b^3*c - 2*a*b^4*c -

2*a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 +

6*a^2*b*c^3 + 2*a*b^2*c^3 - b^3*c^3 + a*b*c^4 - 3*b^2*c^4 -

2*a*c^5 - 3*b*c^5 - a^5*RB - a^4*b*RB + a^3*b^2*RB + a^2*b^3*RB -

a^4*c*RB - 4*a^3*b*c*RB - a^2*b^2*c*RB + 2*a*b^3*c*RB +

2*a^3*c^2*RB + 2*a^2*b*c^2*RB - a*b^2*c^2*RB + b^3*c^2*RB +

2*a^2*c^3*RB + 4*a*b*c^3*RB + b^2*c^3*RB - a*c^4*RB - b*c^4*RB -

c^5*RB)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + 2*a*b*c^2 - 3*b^2*c^2 +

2*c^4 - 2*a^2*c*RC + 2*b^2*c*RC - 2*c^3*RC))}