Pole. Polar. Theorems of Brianchon and Brokard Definition
Theorem 6
\( A\in b \) if and only if \( \angle AB^*O=90^{\circ} \). Analogously
\( B\in a \) if and only if \( \angle BA^*O=90^{\circ} \),
and it remains to notice that according to the basic
properties of inversion we have
\( \angle AB^*O=\angle BA^*O \).
Definition Theorem 7
Let \( C_1 \) and \( D_1 \)
be the intersection points of \( OA \) with \( k \).
Since the inversion preserves the crossratio
and \( \mathcal R(C_1,D_1;A,A^*)=\mathcal R(C_1,D_1;A^*,A) \) we have
\[ \mathcal H(C_1,D_1;A,A^*).\quad\quad\quad\quad\quad (7)\]
Let \( p \) be the line that contains
\( A \) and intersects \( k \) at \( C \) and \( D \).
Let \( E=CC_1\cap DD_1 \), \( F=CD_1\cap DC_1 \). Since \( C_1D_1 \)
is the diameter of \( k \) we have \( C_1F\bot D_1E \) and
\( D_1F\bot C_1E \), hence \( F \) is the orthocenter of
the triangle \( C_1D_1E \). Let \( B=EF\cap CD \) and
\( \bar{A}^*=EF\cap C_1D_1 \). Since
\[ C_1D_1A\bar{A}^* \frac{E}{\overline\wedge} CDAB \frac{F}{\overline\wedge} D_1C_1A\bar{A}^*\]
have \( \mathcal H(C_1,D_1;A,\bar{A}^*) \) and \( \mathcal H(C,D;A,B) \).
(7) now implies two facts:
\( 1^{\circ} \) From \( \mathcal H(C_1,D_1;A,\bar{A}^*) \) and \( \mathcal H(C_1,D_1;A,A^*) \) we get \( A^*=\bar{A}^* \), hence \( A^*\in EF \). However, since \( EF\bot C_1D_1 \), the line \( EF=a \) is the polar of \( A \). \( 2^{\circ} \) For the point \( B \) which belongs to the polar of \( A \) we have \( \mathcal H(C,D;A,B) \). This completes the proof. Theorem 8 (Brianchon)
We will use the convention in which the points will be denoted
by capital latin letters, and their respective polars with
the corresponding lowercase letters.
Denote by \( M_i \), \( i=1,2,\dots,6 \), the points of tangency of \( A_iA_{i+1} \) with \( k \). Since \( m_i=A_iA_{i+1} \), we have \( M_i\in a_i \), \( M_i\in a_{i+1} \), hence \( a_i=M_{i1}M_i \). Let \( b_j=A_jA_{j+3} \), \( j=1,2,3 \). Then \( B_j=a_j\cap a_{j+3}=M_{j1}M_j\cap M_{j+3}M_{j+4} \). We have to prove that there exists a point \( P \) such that \( P\in b_1,b_2,b_3 \), or analogously, that there is a line \( p \) such that \( B_1,B_2,B_3\in p \). In other words we have to prove that the points \( B_1 \), \( B_2 \), \( B_3 \) are collinear. However this immediately follows from the Pascal\( \prime \)s theorem applied to \( M_1M_3M_5M_4M_6M_2 \). From the previous proof we see that the Brianchon\( \prime \)s theorem is obtained from the Pascal\( \prime \)s by replacing all the points with their polars and all lines by theirs poles. Theorem 9 (Brokard)
We will prove that \( EG \) is a polar of
\( F \). Let \( X=EG\cap BC \) and \( Y=EG\cap AD \).

20052019 IMOmath.com  imomath"at"gmail.com  Math rendered by MathJax Home  Olympiads  Book  Training  IMO Results  Forum  Links  About  Contact us 