Welcome to our lesson on the Hand-in-Hand model in plane geometry. This is a fundamental configuration where two congruent triangles share a common vertex, creating symmetrical patterns that appear frequently in geometric problems. The model gets its name because the triangles seem to hold hands at their shared vertex.
The key properties of the Hand-in-Hand model are based on congruent triangles. Triangle ABC is congruent to triangle ADE, sharing vertex A. This means corresponding sides are equal: AB equals AD, AC equals AE, and BC equals DE. Most importantly, the connecting segments BE and CD have special properties - they are equal in length and can be rotated into each other around point A.
The Hand-in-Hand model is fundamentally about rotation transformation. When we rotate triangle ABC around point A by a certain angle, it coincides with triangle ADE. This rotation maps point B to point D, and point C to point E. The rotation angle is determined by the angle between corresponding sides. This transformation property is what makes the connecting segments BE and CD equal in length.
Now let's prove that segments BE and CD are equal. Since triangles ABC and ADE are congruent, we know that AB equals AD, AC equals AE, and angle BAC equals angle DAE. To prove BE equals CD, we consider triangles ABE and ADC. These triangles have AB equal to AD, AE equal to AC, and angle BAE equal to angle DAC. By the SAS congruence criterion, triangles ABE and ADC are congruent, which means BE equals CD.