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This implies 20,02, > SQ, an inequality from which we see that the maximum of |AQRS| is attained when SQ || O1O2. ees “ae 6 Oe Se ae Since the location of the points O,, O2, O3 depends only on AABC and angles a, (6,7, the result just established shows that there exists a unique area-maximizing triangle in Cyg,. Given al area-minimizing triangle ADEF in the class Zag ,, we can associate with it a homotopic triangl AQRS in Cagy, which can be constructed following the recipe given in Proposition 2.1) Using the Remark 1, the latter triangle is area-maximizing in Cyg, and, hence, must coincide with the unique area-maximizing triangle described in the first part of the current proof. This shows t is itself uniquely determined by the property that it is area-minimizing in Z,g,. This concludes the proof of the first claim made in the statement of Theorem[2.4] The above considerations also shov that ADEF and AQRS are homotopic, thus proving item (Z). rr feeN Foy me nr 7 fs 7. 7D p> of I SNnNnni hat ADEF

Figure 4 This implies 20,02, > SQ, an inequality from which we see that the maximum of |AQRS| is attained when SQ || O1O2. ees “ae 6 Oe Se ae Since the location of the points O,, O2, O3 depends only on AABC and angles a, (6,7, the result just established shows that there exists a unique area-maximizing triangle in Cyg,. Given al area-minimizing triangle ADEF in the class Zag ,, we can associate with it a homotopic triangl AQRS in Cagy, which can be constructed following the recipe given in Proposition 2.1) Using the Remark 1, the latter triangle is area-maximizing in Cyg, and, hence, must coincide with the unique area-maximizing triangle described in the first part of the current proof. This shows t is itself uniquely determined by the property that it is area-minimizing in Z,g,. This concludes the proof of the first claim made in the statement of Theorem[2.4] The above considerations also shov that ADEF and AQRS are homotopic, thus proving item (Z). rr feeN Foy me nr 7 fs 7. 7D p> of I SNnNnni hat ADEF

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The parallels through A, B and C to the sides of the triangle A M,N, P, determine the tri- angle M"”N"P” which is homotopic to A M,N,P, and circumscribed to AABC. As a result, A M"N"P" € Cagy which forces
The parallels through A, B and C to the sides of the triangle A M,N, P, determine the tri- angle M"”N"P” which is homotopic to A M,N,P, and circumscribed to AABC. As a result, A M"N"P" € Cagy which forces
Before proceeding with the proof of this theorem, a comment is in order.
Before proceeding with the proof of this theorem, a comment is in order.
If M, M2Mz3 and N;N2N3 are the pedal triangles of M and N respectively, with
If M, M2Mz3 and N;N2N3 are the pedal triangles of M and N respectively, with
Proof. Since A D, EF, is the antipedal triangle of M,, with respect to A ABC, we can obtain (reasoning similarly to what was done in the proof of Theorem [3.1] to show that YM, = ABMA-— X ACB) that Theorem 3.2. The pedal points of the six triangles, Aagy, Aaye, ABay, AByas Ayap; Ayga, that ar area-minimizers in the classes Tagy, Lave, Lgay, Leyva, Lyas, Lyba, respectively, all lie on a circle.
Proof. Since A D, EF, is the antipedal triangle of M,, with respect to A ABC, we can obtain (reasoning similarly to what was done in the proof of Theorem [3.1] to show that YM, = ABMA-— X ACB) that Theorem 3.2. The pedal points of the six triangles, Aagy, Aaye, ABay, AByas Ayap; Ayga, that ar area-minimizers in the classes Tagy, Lave, Lgay, Leyva, Lyas, Lyba, respectively, all lie on a circle.
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