In the given triangle $\triangle ABC$, the following information is provided:

• $\ell(AB) = 6$
• $\ell(BC) = 7$
• $\ell(AC) = 8$
• $AD$ is perpendicular to $BC$
• $AE$ is an angle bisector of $\angle A$
• $AF$ is a median of $\triangle ABC$

Key Definitions:

1. Perpendicular line (AD): This means $AD \perp BC$, so $\triangle ABD$ and $\triangle ADC$ are right triangles.
2. Angle bisector (AE): Since $AE$ bisects $\angle A$, it divides $\angle BAC$ into two equal angles. Also, by the Angle Bisector Theorem, the sides opposite these angles have a specific ratio.
3. Median (AF): $AF$ is the median, so $F$ is the midpoint of $BC$. Therefore, $BF = FC = \frac{7}{2} = 3.5$.

Tasks:

From this setup, you can explore properties related to:

• Area of the triangle: Using the given sides and the perpendicular, you can find the height and apply the area formula for a triangle.
• Right triangle relationships: Apply the Pythagorean theorem in $\triangle ABD$ and $\triangle ADC$.
• Angle Bisector Theorem: This can help in finding relations between the sides of the triangle.
• Coordinate geometry: Assigning coordinates to the vertices of the triangle might be useful for further calculations.

Would you like to explore a specific aspect or solve for something specific? Here are some questions to guide further exploration:

1. What is the length of the height $AD$ using the Pythagorean theorem?
2. Can you calculate the area of $\triangle ABC$ using $AD$ as the height?
3. What is the length of $AE$ using the Angle Bisector Theorem?
4. How can you use the median $AF$ to explore the balance between sides $AB$ and $AC$?
5. What are the internal angles of $\triangle ABC$, and how do the bisectors divide them?

Tip:

Remember that the properties of perpendiculars, angle bisectors, and medians divide the triangle in various meaningful ways that can help simplify calculations.