Triangle Proportionality Theorem
Verify AD/DB=AE/EC via triangle proportionality theorem. Find missing segment when parallel. Free geometry calculator with step-by-step results for student.
Enter any 3 values to find the missing segment, or all 4 to check proportionality.
About This Calculator
The Triangle Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. This calculator helps geometry students verify the theorem by checking if AD/DB equals AE/EC or find a missing segment length when three out of four values are known. Whether you are studying for a geometry exam or teaching the concept of proportional segments in triangles, this tool provides instant and accurate results.
The calculator uses the core relationship AD/DB = AE/EC, where DE is a line parallel to side BC of triangle ABC, intersecting AB at D and AC at E. Given any three of the four segment lengths, the calculator solves for the fourth using cross-multiplication. For example, if AD, DB, and AE are known, EC is computed as (AE x DB) / AD. If all four values are entered, the calculator checks whether the two ratios are equal within a tolerance of 0.001, confirming whether the line is parallel to the base.
The Triangle Proportionality Theorem is also known as Thales' theorem or the Basic Proportionality Theorem in some geometry curricula. It forms the foundation for understanding similarity in triangles and is widely used in construction, engineering, and architectural design where parallel lines and proportional measurements are essential. Understanding this theorem helps students grasp more advanced concepts such as the Midpoint Theorem, similarity proofs, and coordinate geometry applications.
Frequently Asked Questions
What is the triangle proportionality theorem?
The triangle proportionality theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those two sides proportionally. In triangle ABC, if DE is parallel to BC (with D on AB and E on AC), then AD/DB = AE/EC.
How do I use this triangle proportionality theorem calculator?
Enter any three of the four segment lengths AD, DB, AE, EC and click Calculate. The tool will find the missing segment length using the proportion AD/DB = AE/EC. If you enter all four values, it will verify whether the ratios are equal.
What is the converse of the triangle proportionality theorem?
If a line divides two sides of a triangle proportionally, then that line must be parallel to the third side. In other words, if AD/DB = AE/EC, then DE must be parallel to BC.
How do you prove the triangle proportionality theorem?
If DE is parallel to BC, then angle ADE is congruent to angle ABC and angle AED is congruent to angle ACB (corresponding angles). By AA similarity, triangle ABC is similar to triangle ADE. Therefore AB/AD = AC/AE. Using the denominator subtraction property, (AB minus AD)/AD = (AC minus AE)/AE, which gives DB/AD = EC/AE. Rearranging yields AD/DB = AE/EC.
What are common mistakes when applying the triangle proportionality theorem?
A common mistake is misidentifying which sides are divided. The parallel line must intersect the other two sides at distinct points. Another mistake is reversing the ratio: ensure AD corresponds to AE and DB corresponds to EC, not the other way around. Also verify that the line is truly parallel to the third side.
Can I use this calculator for any type of triangle?
Yes, the triangle proportionality theorem applies to any triangle -- acute, right, or obtuse -- as long as the line is drawn parallel to one of the sides. This calculator works for all triangle types regardless of their shape or angles.
What units should I use for segment lengths?
Any unit of length works, including centimeters, inches, meters, or feet, as long as all four segments use the same unit. The calculator uses raw numbers, so unit consistency matters more than the specific unit chosen.
Is this triangle proportionality theorem calculator free?
Yes, all calculators on Calculy including this one are completely free to use with no registration, download, or subscription required.