We are tasked with solving for the angle $\angle EFG$ in the triangle $ABC$, given that:

• $\angle BAC = 66^\circ$,
• $\angle ABC = 72^\circ$,
• $E$, $F$, and $G$ are the midpoints of segments $BH$, $BC$, and $AH$, respectively,
• $H$ is the orthocenter of $\triangle ABC$.

Step 1: Determine $\angle BCA$

The sum of internal angles of a triangle is $180^\circ$. Therefore, for triangle $ABC$:

$\angle BAC + \angle ABC + \angle BCA = 180^\circ$

Substitute the given values:

$66^\circ + 72^\circ + \angle BCA = 180^\circ$

$\angle BCA = 180^\circ - 138^\circ = 42^\circ$

Thus, $\angle BCA = 42^\circ$.

Step 2: Key properties of $E$, $F$, and $G$

• $H$ is the orthocenter, so segments $AH$, $BH$, and $CH$ are altitudes of the triangle.
• $E$ is the midpoint of $BH$,
• $F$ is the midpoint of $BC$,
• $G$ is the midpoint of $AH$.

By the midline theorem, segments $EF$, $FG$, and $GE$ are parallel to the sides of the triangle $ABC$ and half their lengths.

Step 3: Parallel properties and angle relationships

Since $EF \parallel AC$, $FG \parallel AB$, and $GE \parallel BC$, triangle $EFG$ is similar to triangle $ABC$.

The angles in $\triangle EFG$ will be equal to the corresponding angles in $\triangle ABC$:

$\angle EFG = \angle BAC = 66^\circ$

Final Answer:

$\angle EFG = 66^\circ$

Do you need further clarification or a detailed explanation? Let me know!

Here are 5 related questions you might find useful:

1. How do you determine the orthocenter of a triangle geometrically?
2. What are the properties of midline segments in a triangle?
3. How can you prove two triangles are similar using parallel lines?
4. What are the key properties of medians, altitudes, and midpoints in a triangle?
5. How do you calculate unknown angles in any triangle given two angles?

Tip:

Always check for parallel lines in geometric problems—they often simplify angle calculations through similarity or congruence.