Lesson Altitudes in an isosceles triangle

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<H2>Altitudes in an isosceles triangle</H2>

It is better to read this lesson after the lessons <A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A> and <A HREF=http://www.algebra.com/algebra/homework/Triangles/Isosceles-triangles.lesson> Isosceles triangles</A> 
that are under the topic Triangles in the section Geometry in this site.

Theorem 1
If in a triangle the two altitudes are of equal length, then the triangle is isosceles.

<TABLE>
 <TR>
 <TD>
Proof

Let ABC be a triangle with altitudes AD and BE of equal length (Figure 1).&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
We need to prove that the sides AC and BC are of equal length.

Consider the triangles ADC and BEC.
They are the right triangles with the common angle ACB.
The angles CAD and CBE are congruent as the complementary angles to the angle ACB.
Thus, the triangles ADC and BEC have congruent sides AD and BE,
as well as congruent angles ADC and BEC (right angles) and congruent 
angles CAD and CBE.
Therefore, the triangles ADC and BEC are congruent, in accordance 
to the postulate P2 (ASA) (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A> of the 
topic Triangles in the section Geometry in this site).
Hence, the straight segments AC and BC are of equal length as the corresponding 
sides of these triangles.

The proof is completed.
 </TD>
 <TD>
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 line( 0.3, 0.5, 3.7, 0.5), 
 line( 0.3, 0.5, 2.0, 4.5),
 line( 2.0, 4.5, 3.7, 0.5),

locate(0.3, 0.5, A),
            locate(3.7, 0.5, B),
            locate(2.0, 4.9, C),

line (0.3, 0.5, 3.2, 1.7),
            line (3.7, 0.5, 0.8, 1.7),

locate(3.3, 1.9, D),
            locate(0.6, 1.9, E),

arc (3.2, 1.7, 0.8, 0.8, 155, 240),
            arc (0.8, 1.7, 0.8, 0.8, 300,  25),

arc (2.0, 4.5, 0.8, 0.8, 65, 115),
            arc (2.0, 4.5, 1.0, 1.0, 65, 115),

arc (0.3, 0.5, 0.8, 0.8, 295, 335),
            arc (0.3, 0.5, 0.9, 0.9, 295, 335),
            arc (0.3, 0.5, 1.0, 1.0, 295, 335),

arc (3.7, 0.5, 0.8, 0.8, 205, 245),
 arc (3.7, 0.5, 0.9, 0.9, 205, 245),
 arc (3.7, 0.5, 1.0, 1.0, 205, 245)
)}}}
Figure 1. To the Theorem 1
 </TD>
 </TR>
</TABLE>

The opposite statement to the Theorem 1 is true also:

Theorem 2
If a triangle is isosceles, then the two altitudes drawn from vertices at the base to the sides are of equal length.

<TABLE>
 <TR>
 <TD>
Proof
Let ABC be an isosceles triangle with sides AC and BC of equal length (Figure 2).&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
We need to prove that the altitudes AD and BE are of equal length.

Consider the triangles ADB and AEB. 
They are the right triangles.
The angles EAB and ABD are congruent as the triangle ABC is isosceles (see 
the lesson <A HREF=> Isosceles triangles</A> of the topic Triangles in the section Geometry in this site).
If so, the angles EBA and DAB are congruent as the complementary angles
to the angles EAB and ABD.
Thus, the triangles ADB and AEB have the common side AB, as well as 
congruent angles EAB and ABD and congruent angles EBA and DAB.
Therefore, the triangles ADB and AEB are congruent, in accordance 
to the postulate P2 (ASA) (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A> of the 
topic Triangles in the section Geometry in this site).
Hence, the altitudes AD and BE are of equal length as the corresponding 
sides of these triangles.

The proof is completed.
 </TD>
 <TD>
{{{drawing( 200, 250, 0, 4, 0, 5, 
 line( 0.3, 0.5, 3.7, 0.5), 
 line( 0.3, 0.5, 2.0, 4.5),
 line( 2.0, 4.5, 3.7, 0.5),

locate(0.3, 0.5, A),
            locate(3.7, 0.5, B),
            locate(2.0, 4.9, C),

line (0.3, 0.5, 3.2, 1.7),
            line (3.7, 0.5, 0.8, 1.7),

locate(3.3, 1.9, D),
            locate(0.6, 1.9, E),

arc (3.2, 1.7, 0.6, 0.6, 70, 155),
            arc (0.8, 1.7, 0.6, 0.6, 30, 115),

arc (0.3, 0.5, 0.6, 0.6, 300, 360),
            arc (0.3, 0.5, 0.8, 0.8, 300, 360),

arc (3.7, 0.5, 0.6, 0.6, 180, 240),
            arc (3.7, 0.5, 0.8, 0.8, 180, 240),

arc (0.3, 0.5, 1.2, 1.2, 335, 360),
            arc (0.3, 0.5, 1.3, 1.3, 335, 360),
            arc (0.3, 0.5, 1.4, 1.4, 335, 360),

arc (3.7, 0.5, 1.2, 1.2, 180, 205),
 arc (3.7, 0.5, 1.3, 1.3, 180, 205),
 arc (3.7, 0.5, 1.4, 1.4, 180, 205)
)}}}
Figure 2. To the Theorem 2
 </TD>
 </TR>
</TABLE>

Summary

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

Try to prove yourself similar statements for medians and angle bisectors of the isosceles triangle:

A triangle is isosceles if and only if the two medians 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 altitudes of the current lesson.
In any case, you can find the full proofs in the lessons 
<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Medians-in-an-isosceles-triangle.lesson> Medians in an isosceles triangle</A> and <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Angle-bisectors-in-an-isosceles-triangle.lesson> Angle bisectors in an isosceles triangle</A> 
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.

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson>Congruence tests for triangles</A>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;under the topic Triangles in the section Geometry;

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Isosceles-triangles.lesson>Isosceles triangles</A>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;under the topic Triangles in the section Geometry;

<A HREF=http://www.algebra.com/algebra/homework/Triangles/An-altitude-a-median-and-an-angle-bisector-in-the-isosceles-triangle.lesson>An altitude, a median and an angle bisector in the isosceles triangle</A> &nbsp;under the topic Triangles in the section Geometry;

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Medians-in-an-isosceles-triangle.lesson>Medians in an isosceles triangle</A>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;under the topic Geometry in the section Word problems;

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Angle-bisectors-in-an-isosceles-triangle.lesson> Angle bisectors in an isosceles triangle</A>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;under the topic Geometry in the section Word problems.