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FIGURE 10. A beautiful sub-case of the Liang-Zelich Theorem. Theorem 3.1. Let P be a point on the Euler line of a triangle ABC. Suppose iy = t (this ratio is directed), where O, H denote the circumcenter and orthocenter of ABC. Let the antipedal triangle of P wrt ABC be A*B*C* and let P* be a point on the Euler line of A* B*C* such that io = t, where O*, H*™ are defined analogously. Then P* is also on the Euler line of ABC.

Figure 10 A beautiful sub-case of the Liang-Zelich Theorem. Theorem 3.1. Let P be a point on the Euler line of a triangle ABC. Suppose iy = t (this ratio is directed), where O, H denote the circumcenter and orthocenter of ABC. Let the antipedal triangle of P wrt ABC be A*B*C* and let P* be a point on the Euler line of A* B*C* such that io = t, where O*, H*™ are defined analogously. Then P* is also on the Euler line of ABC.

Related Figures (10)

Ficure 1. A fundamental result on rectangular hyperbolas. Proposition 1.1. Consider an arbitrary triangle ABC on a rectangular hyperbola. Then the orthocenter of ABC is also on the rectangular hyperbola. The converse is also true.
Ficure 1. A fundamental result on rectangular hyperbolas. Proposition 1.1. Consider an arbitrary triangle ABC on a rectangular hyperbola. Then the orthocenter of ABC is also on the rectangular hyperbola. The converse is also true.
FIGURE 2. Sondat’s Theorem - a little-known theorem. Lemma 1.3. Let ABC and A'B'C" be two fixed triangles that are not homothetic. Let P be a point such that the parallels through A’, B’,C’ to AP, BP,CP respectively concur at a point Q. Then the locus of points P is either a circumconic of ABC or the line at infinity.
FIGURE 2. Sondat’s Theorem - a little-known theorem. Lemma 1.3. Let ABC and A'B'C" be two fixed triangles that are not homothetic. Let P be a point such that the parallels through A’, B’,C’ to AP, BP,CP respectively concur at a point Q. Then the locus of points P is either a circumconic of ABC or the line at infinity.
FIGURE 3. An interesting involutive transformation commuting to isogonal conjugacy.
FIGURE 3. An interesting involutive transformation commuting to isogonal conjugacy.
Figure 4. A simple yet effective configuration.
Figure 4. A simple yet effective configuration.
FIGURE 5. The Liang-Zelich Theorem.
FIGURE 5. The Liang-Zelich Theorem.
Theorem 2.3. Given triangle ABC and a point P, let Hy,Hp,Ho denote the orthocenters of PBC,PAC,PAB. Consider a triangle DEF orthologic and per- spective with ABC such that Oapc(DEF) = P,Opgr(ABC) = P’. The following properties are true: FIGURE 6. An amazing novel Theorem on a triple of orthologic triangles.
Theorem 2.3. Given triangle ABC and a point P, let Hy,Hp,Ho denote the orthocenters of PBC,PAC,PAB. Consider a triangle DEF orthologic and per- spective with ABC such that Oapc(DEF) = P,Opgr(ABC) = P’. The following properties are true: FIGURE 6. An amazing novel Theorem on a triple of orthologic triangles.
FIGURE 7. First proof of the Assertion 1. 2.2. Properties of the t(P, ABC) ratio. In this subsection, we present another way to prove Assertion 1 using nothing but Proposition 1.6. The proof is special in that it gives us a crucial corollary that will help prove Assertion 3.
FIGURE 7. First proof of the Assertion 1. 2.2. Properties of the t(P, ABC) ratio. In this subsection, we present another way to prove Assertion 1 using nothing but Proposition 1.6. The proof is special in that it gives us a crucial corollary that will help prove Assertion 3.
FIGURE 8. Diagram 1 for the second proof of the Assertion 1.
FIGURE 8. Diagram 1 for the second proof of the Assertion 1.
FIGURE 9. Diagram 2 for the second proof of the Assertion 1.
FIGURE 9. Diagram 2 for the second proof of the Assertion 1.
Problem 3.2. Let ABC be a triangle and O its circumcenter. The perpendicular bisectors of BC and AO intersect at Og. Define Oy,O~ similarly. What point represents the intersection of the Euler lines of O,O,O, and ABC? We will prove a theorem before presenting a proof of this problem.
Problem 3.2. Let ABC be a triangle and O its circumcenter. The perpendicular bisectors of BC and AO intersect at Og. Define Oy,O~ similarly. What point represents the intersection of the Euler lines of O,O,O, and ABC? We will prove a theorem before presenting a proof of this problem.
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