Notes on Euclid and Circle Geometry

Introduction to Euclid

• Euclid was a Greek mathematician around 2400 years ago.
• Known as the father of geometry, specifically Euclidean geometry.
• He organized and summarized geometric theories for future use.

Circle Geometry Theorems by Euclid

• Objective: Understand basic circle geometry using Euclid's discoveries.

Drawing a Circle and a Chord

• Draw a circle and locate its center point.
• Draw a chord: a line that touches two sides of the circle but does not pass through the center.
• Draw a line from the center to the midpoint of the chord.

Theorem 1: Perpendicular Line from Center to Midpoint

• When a line from the center hits the midpoint of any chord, it forms a 90-degree angle.
• Example: Angle OBC = 90 degrees if line OB is to the midpoint of chord BC.
• Reason: A line from the center to the midpoint of a chord always creates a right angle (90 degrees).

Theorem 2: Equal Segments from Perpendicular Line

• If a line from the center forms a 90-degree angle with a chord, it divides the chord into two equal lengths.
• Example: If AB = BC, then the line from the center is perpendicular to chord AB.
• Reason: A perpendicular line from the center ensures the chord is divided equally.

Application of Theorems

• First Theorem Application: Given two equal chord segments, the angle must be 90 degrees if the line is from the center to the midpoint.
• Second Theorem Application: If it is known the angle is 90 degrees, the chord segments are equal.

Practice Instructions

• Practice by drawing circles, chords, and lines as described.
• Verify Euclid’s theorems by measuring angles and lengths.

Additional Notes

• Discussion on different spellings of "center" (centre vs. center).

By practicing these geometrical constructions, students will see the accuracy of Euclid's findings, which remain true in all cases as described.