Lesson Medians in an isosceles triangle

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Medians in an isosceles triangle

It is better to read this lesson after the lessons Congruence tests for triangles and Isosceles triangles
that are under the topic Triangles in the section Geometry in this site.

Theorem 1
If a triangle is isosceles, then the two medians drawn from vertices at the base to the sides are of equal length.
 Proof Let ABC be an isosceles triangle with sides AC and BC of equal length (Figure 1).                 We need to prove that the medians AD and BE are of equal length. Consider the triangles ADC and BEC. They have two congruent sides that include congruent angles. Indeed, AC = BC by the condition, because the triangle ABC is isosceles. Since the lateral sides AC and BC are of equal length, their halves EC and DC are of equal length too: EC = DC. Finally, the angle ECD is the common angle. Thus, the triangles ADC and BEC are congruent, in accordance to the postulate P1 (SAS) (see the lesson Congruence tests for triangles of the topic Triangles in the section Geometry in this site). Hence, the medians AD and BE are of equal length as the corresponding sides of these triangles. The proof is completed. Figure 1. To the Theorem 1

The opposite statement to the Theorem 1 is true also.

Theorem 2
If in a triangle the two medians drawn from vertices at the base to the sides are of equal length, then the triangle is isosceles.
 Proof Let ABC be a triangle with medians AD and BE of equal length (Figure 2).         We need to prove that the sides AC and BC are of equal length. Connect the points E and D by the straight line segment ED (Figure 3). Since the points E and D are midpoints of the sides AC and BC, the straight line ED is parallel to the triangle side AB. It is proved in the lesson The line segment joining the midpoints of two sides of a triangle (under the topic Triangles in the section Geometry in this site). Draw the straight line from the point E parallel to the median AD till the intersection with the continuation of the straight line AB (Figure 3). Mark the intersection point as F. Also connect the points F and A. Since in the quadrilateral FADE the opposite sides FA and ED are parallel and the opposite sides FE and AD are parallel too, the segment FE Figure 2. To the Theorem 2 Figure 3. To the proof of the Theorem 2
is of the same length as the median AD. It is proved in the lesson Properties of the sides of parallelograms under the topic Triangles in the section Geometry in this site.

This means that the segment FE is of the same length as BE, because the medians AD and BE are of equal length.
So, the triangle FBE is isosceles and its angles BFE and FBE are congruent.
This implies that the angles BAD and ABE are congruent, because the angles BFE and BAD are congruent as the corresponding angles at parallel straight lines FE and AD (see the lesson Parallel lines under the topic Angles, complementary, supplementary angles in the section Geometry in this site).

Now, you have two triangles ABE and BAD with the common base AB, with the congruent lateral sides AD and BE, and with the congruent angles BAD and ABE that are included between congruent sides. In accordance to the postulate P1 (SAS) of the lesson Congruence tests for triangles (under the topic Triangles in the section Geometry in this site), these triangles are congruent.
This implies that the corresponding sides AE and BD are congruent, and hence the lateral sides AC and BC are of equal length, because their lengths are twice the lengths of the segments AE and BD. The proof is completed.

Summary
A triangle is isosceles if and only if the two medians drawn from vertices at the base to the sides are of equal length.

Similar statements are valid for altitudes and angle bisectors of the isosceles triangle:
A triangle is isosceles if and only if the two altitudes drawn from vertices at the base to the sides are of equal length.
If a triangle is isosceles, then two angle bisectors drawn from vertices at the base to the sides are of equal length.

The proofs are similar to that for the medians of the current lesson.
Try to prove these statements yourself.
In any case, you can find the full proofs in the lessons
Altitudes in an isosceles triangle and Angle bisectors in an isosceles triangle
under the current topic (the topic Geometry in the section Word problems).

For your convenience, below is the list of my relevant lessons in this site in the logical order.
Congruence tests for triangles                                                      under the topic Triangles in the section Geometry;
Isosceles triangles                                                                           under the topic Triangles in the section Geometry;
An altitude, a median and an angle bisector in the isosceles triangle under the topic Triangles in the section Geometry;
Altitudes in an isosceles triangle                                                   under the topic Geometry in the section Word problems;
The line segment joining the midpoints of two sides of a triangle   under the topic Triangles in the section Geometry       (the auxiliary material to the present lesson);
Properties of the sides of parallelograms                                          under the topic Triangles in the section Geometry        (the auxiliary material to the present lesson);
Angle bisectors in an isosceles triangle                                           under the topic Geometry in the section Word problems.

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