Dear Jeff and Hauke,

In a triangle ABC we can inscribe infinitely

many triangles A1B1C1 that are similar to

a given triangle A'B'C'. The method is the following:

In message #16970 we have seen how to construct the

unique point P such that the pedal triangle XYZ of P

is similar to A'B'C'.

The line PX rotated by an angle w meets BC at the point A1.

The line PY rotated by an angle w meets CA at the point B1.

The line PZ rotated by an angle w meets AB at the point C1.

The triangle A1B1C1 is similar to A'B'C' and w is arbitrary.

If A1B1C1 is a cevian then tan(w) must be the root of

a 3-degree equation. Hence there are 3 such points.

If the triangle A'B'C' is equilateral then P is not unique.

There are two such points X(15), X(16).

If Q is the isogonal conjugate of P then the antipedal triangle

X'Y'Z' of Q wrt ABC is a triangle circumscribed to ABC and

similar to A'B'C'. As previously the rotated QA meets Y'Z' at

a point A2. Similarly we find B2, C2 and the triangle A2B2C2

is a triangle similar to ABC and X'Y'Z' is circumscribed to A2B2C2.

If A'B'C' is an equilateral triangle the points Q are the points

X(13) and X(14).

We can find that the triangles A2B2C2 and X'Y'Z' are

perspective iff tan(w) is the root of a 3-degree equation.

Hence there are 6 points such that Hauke wants to find

and the barycentrics must be complicated.

Best regards

Nikos Dergiades

> Dear Nikos and friends,

>

> Sorry, I meant 'inscribed' triangle

> A'B'C'. Also, I meant similar

> triangles A'B'C'.

> Sincerely, Jeff

> > Dear Nikos,

> >

> > In message #16970 you gave a proof how to construct a

> triangle A'B'C'

> > inscrided in a given triangle ABC. But now, how to

> construct A'B'C'

> > circumscribed to ABC?

> >

> > Sincerely, Jeff

> Circumscribe an equilateral triangle A'B'C'

> around ABC

> such that ABC is a Cevian triangle of A'B'C'

> with

> respect to some point O (which coordinates I desperately

> try to compute for the Kimberling triangle but for each

> of my NSolve[] tries I get different results. :-O )

>

> Hauke

>

> Yahoo! Groups Links

>

>

>

___________________________________________________________

Χρησιμοποιείτε Yahoo!;

Βαρεθήκατε τα ενοχλητικά μηνύματα (spam); Το Yahoo! Mail

διαθέτει την καλύτερη δυνατή προστασία κατά των ενοχλητικών

μηνυμάτων

http://login.yahoo.com/config/mail?.intl=gr