Enter a triangle side length to compute its parallel midsegment
Side BC (third side)
Side AB
Side AC
Result
DE (mid of AB & AC)
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DF (mid of AB & BC)
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EF (mid of AC & BC)
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Detailed Derivation
Midsegment Theorem Formula
DE = BC / 2, DE is parallel to BC
DF = AC / 2, DF is parallel to AC
EF = AB / 2, EF is parallel to AB
The Triangle Midsegment Theorem is a fundamental result in geometry. The segment joining the midpoints of any two sides is parallel to the third side and exactly half its length.
⚠Side lengths must satisfy the triangle inequality. All three sides must form a valid triangle.
What Is the Midsegment Theorem?
The Triangle Midsegment Theorem states that a segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. This theorem is used extensively in geometry proofs and construction.
Parallel Property
DE is parallel to BC. This can be proven using corresponding angles or coordinate geometry.
Half-Length Property
DE = BC/2. This is the key computational relationship used to find unknown lengths.
Medial Triangle
Connecting all three midpoints creates the medial triangle, which is similar to the original with a 1:2 ratio.
Area Relationship
The medial triangle has exactly one-quarter of the original triangle area. Four copies of the medial triangle fill the original.
Teaching Example: Triangle with sides AB=8, AC=12, BC=10. Midsegment DE connecting midpoints of AB and AC: DE = BC/2 = 5, DE parallel to BC. DF = AC/2 = 6, EF = AB/2 = 4. The medial triangle has perimeter = 5+6+4 = 15 (half of original perimeter 30).
The segment connecting midpoints of two sides is parallel to the third side and half its length. DE = BC/2 and DE is parallel to BC.
How many midsegments does a triangle have?▼
Three midsegments. They connect midpoints of each pair of sides, forming the medial triangle which is similar to the original.
What is the area of the medial triangle?▼
The medial triangle area equals one-quarter of the original triangle area. Its perimeter equals half of the original perimeter.
Is the midsegment always parallel?▼
Yes, by the Midsegment Theorem, the segment is always parallel to the third side. This can be proven using corresponding angles from parallel line properties.
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