triangle ABC has D a point on side BC. Points E and F are on sides AB and AC such that DE and DF are the angle bisectors of angle ADB and angle ADC respectively. Find the value of (AE)/(EB) cdot (BD)/(DC) cdot (CF)/(FA) .

Liberty Page 2022-09-23 Answered
$\mathrm{△}ABC$ has D a point on side BC. Points E and F are on sides AB and AC such that DE and DF are the angle bisectors of $\mathrm{\angle }ADB$ and $\mathrm{\angle }ADC$ respectively. Find the value of $\frac{AE}{EB}\ast \frac{BD}{DC}\ast \frac{CF}{FA}$
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Mckayla Carlson
Step 1
In $\mathrm{△}ADB$, angle bisector DE divides AC in ratio of adjacent sides:
$\frac{AE}{EB}=\frac{AD}{DB}$
Step 2
Similarly $\mathrm{△}ACD$:
$\frac{CF}{AF}=\frac{CD}{AD}$
Therefore $\frac{BD}{DC}\cdot \frac{AE}{EB}\cdot \frac{CF}{FA}=\frac{BD}{DC}\cdot \frac{AD}{BD}\cdot \frac{CD}{AD}=1$