O proprietate interesanta a bisectoarei.

[Virgil Nicula]

Fiind dat un triunghi \( ABC \) notam piciorul \( D\in (BC) \) al bisectoarei interioare din varful \( A \)

si pentru un punct \( P\in (AD) \) notam intersectiile \( E\in BP\cap AC \) , \( F\in CP\cap AB \) si

\( X\in BE\cap DF \) , \( Y\in CF\cap DE \) . Sa se arate ca \( [AD \) este bisectoarea unghiului \( \widehat {XAY} \) .

Generalizare.

Fiind dat \( \triangle ABC \) si un punct interior \( P \) notam \( \left\|\begin{array}{c}
D\in AP\cap BC\\\\\\\\
E\in BP\cap AC\\\\\\\\
F\in CP\cap AB\end{array}\right\| \) , \( \left\|\begin{array}{c}
X\in BE\cap DF\\\\\\\\
Y\in CF\cap DE\end{array}\right\| \)

si \( \left\|\begin{array}{ccc}
m(\angle BAX)=x & ; & m(\angle CAY)=y\\\\\\\\
m(\angle PAX)=u & ; & m(\angle PAY)=v\end{array}\right\| \) . Sa se arate ca \( \frac {\sin (x+u)}{\sin (y+v)}=\frac {\sin x\sin v}{\sin y\sin u} \) .

[Mateescu Constantin]

Generalizare.

Fiind dat \( \triangle ABC \) si un punct interior \( P \) notam \( \left\|\ \begin{array}{ccc}
D & \in & AP & \cap & BC\\\\\\\\
E & \in & BP & \cap & AC\\\\\\\\
F & \in & CP & \cap & AB\ \end{array}\right\| \) , \( \left\|\ \begin{array}{cc}
X & \in & BE & \cap & DF\\\\\\\\
Y & \in & CF & \cap & DE\ \end{array}\right\| \)

si \( \left\|\ \begin{array}{ccc}
m(\angle BAX) & = & x & ; & m(\angle CAY) & = & y\\\\\\\\
m(\angle PAX) & = & u & ; & m(\angle PAY) & = & v\ \end{array}\right\| \) . Sa se arate ca \( \frac {\sin (x+u)}{\sin (y+v)}=\frac {\sin x\sin v}{\sin y\sin u} \) .

\( \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)

Demonstratie : Aplicam teorema lui Ceva sub forma trigonometrica pentru punctul / triunghiul:

\( \left\| \begin{array}{cc} X/\triangle ABD & : & \frac{\sin x}{\sin u} & \cdot & \frac{\sin(\angle ADF)}{\sin(\angle BDF)} & \cdot & \frac{\sin(\angle DBP)}{\sin(\angle PBA)} & = & 1 & \Longleftrightarrow & \frac{\sin x}{\sin u} & = & \frac{\sin(\angle BDF)}{\sin(\angle ADF)} & \cdot & \frac{\sin(\angle ABE)}{\sin(\angle CBE)} && (1) & \\\\\\\\
Y/\triangle ADC & : & \frac{\sin v}{\sin y} & \cdot & \frac{\sin(\angle ACP)}{\sin(\angle DCP)} & \cdot & \frac{\sin(\angle CDE)}{\sin(\angle EDA)} & = & 1 & \Longleftrightarrow & \frac{\sin v}{\sin y} & = & \frac{\sin(\angle BCF)}{\sin(\angle ACF)} & \cdot & \frac{\sin(\angle ADE)}{\sin(\angle CDE)} && (2) & \\\\\\\\
P/\triangle ABC & : & \frac{\sin(\angle ABE)}{\sin(\angle CBE)} & \cdot & \frac{\sin(\angle BCF)}{\sin(\angle FCA)} & \cdot & \frac{\sin(\angle CAD)}{\sin(\angle BAD)} & = & 1 & \Longleftrightarrow & \frac{\sin(x+u)}{\sin(v+y)}& = & \frac{\sin(\angle ABE)}{\sin(\angle CBE)} & \cdot & \frac{\sin(\angle BCF)}{\sin(\angle FCA)} && (3) & \ \end{array}\right\| \)

\( \Longrightarrow^{(1)\ \odot\ (2)}\ \begin{array} \frac{\sin x}{\sin u} & \cdot & \frac{\sin v}{\sin y} & = & \frac{\sin(\angle ABE)}{\sin(\angle CBE)} & \cdot & \frac{\sin(\angle BCF)}{\sin(\angle ACF)} & \cdot & \frac{\sin(\angle BDF)}{\sin(\angle ADF)} & \cdot & \frac{\sin(\angle ADE)}{\sin(\angle CDE)}\end{array} \)

\( \ \ \Longleftrightarrow^{(3)}\ \ \ \ \begin{array} \frac{\sin x}{\sin u} & \cdot & \frac{\sin v}{\sin y} & = & \frac{\sin(x+u)}{\sin(v+y)} & \cdot & \frac{\sin(\angle BDF)}{\sin(\angle ADF)} & \cdot & \frac{\sin(\angle ADE)}{\sin(\angle CDE)} \end{array}\ \ \ (4) \)

Pe de alta parte avem : \( \left\|\ \begin{array}{ccc} \frac{BF}{FA} & = & \frac{BD}{AD} & \cdot & \frac{\sin(\angle BDF)}{\sin(\angle ADF)} \\\\\\\\
\frac{AE}{EC} & = & \frac{AD}{DC} & \cdot & \frac{\sin(\angle ADE)}{\sin(\angle CDE)} \end{array} \right|\ \bigodot\ \Longrightarrow\ \begin{array} \frac{BF}{FA} & \cdot & \frac{AE}{EC} & = & \frac{BD}{DC} & \cdot & \frac{\sin(\angle BDF)}{\sin(\angle ADF)} & \cdot & \frac{\sin(\angle ADE)}{\sin(\angle CDE)} \end{array} \)

\( \begin{array}{ccc}\Longleftrightarrow^{\mbox{Ceva}} &\ \frac{\sin(\angle BDF)}{\sin(\angle ADF)} & \cdot & \frac{\sin(\angle ADE)}{\sin(\angle CDE)} & = & 1 \end{array}\ (5) \) . Asadar, din relatiile \( (4) \) si \( (5)\ \Longrightarrow\ \begin{array} \fbox{\ \frac{\sin x}{\sin u} \ \cdot \ \frac{\sin v}{\sin y} \ = \ \frac{\sin(x+u)}{\sin(v+y)}\ }\end{array} \)