Key Principle

A tangent line is PERPENDICULAR to the radius at the point of tangency

Method 1: Tangent at a Point ON the Circle

Steps:

Step 1: Draw the circle with center O and mark point P on the circle

Step 2: Draw radius OP (from center to point P)

Step 3: Extend the radius OP beyond point P

Step 4: Construct a perpendicular line to OP at point P:

• Place compass at P and mark equal arcs on both sides of P on line OP

• From these two points, draw arcs above (or below) P that intersect

• Draw a line through P and the intersection point

✓ This perpendicular line is the tangent to the circle at point P

Method 2: Tangent from an EXTERNAL Point

Steps:

Step 1: Draw circle with center O and mark external point P (outside the circle)

Step 2: Draw line segment OP connecting the external point to the center

Step 3: Find the midpoint M of segment OP using perpendicular bisector construction

Step 4: Using M as center and MO (or MP) as radius, draw an arc that intersects the original circle at two points T₁ and T₂

Step 5: Draw lines PT₁ and PT₂

✓ These two lines are tangent to the circle from external point P

Key Property

Side length of inscribed regular hexagon = Radius of circle

Each central angle = 360° ÷ 6 = 60°

Construction Steps

Step 1: Draw a circle with center O and any radius r

Step 2: Mark any point A on the circle

Step 3: Without changing the compass width (keep radius = r), place compass at point A

Step 4: Draw an arc that intersects the circle; mark this point B

Step 5: Move compass to point B and repeat, marking point C

Step 6: Continue this process to mark points D, E, and F around the circle

Step 7: Connect consecutive points: AB, BC, CD, DE, EF, FA

✓ ABCDEF is a regular hexagon inscribed in the circle

Key Property

An equilateral triangle uses every OTHER vertex of a hexagon

Each central angle = 360° ÷ 3 = 120°

Method 1: Using Hexagon Construction

Step 1: Construct a regular hexagon inscribed in the circle (follow hexagon steps above)

Step 2: Label the six vertices as A, B, C, D, E, F

Step 3: Connect every other vertex: A to C, C to E, E to A

✓ Triangle ACE is an equilateral triangle inscribed in the circle

Method 2: Direct Construction

Step 1: Draw a circle with center O

Step 2: Draw a diameter AB

Step 3: Construct a perpendicular bisector of AB, which passes through O

Step 4: Mark point C where the perpendicular intersects the circle

Step 5: Connect A to C and B to C

✓ Triangle ABC is an equilateral triangle (approximately) — for exact, use Method 1

Key Property

Side length of inscribed square = r√2

r = radius of circle

Each central angle = 360° ÷ 4 = 90°

Construction Steps

Step 1: Draw a circle with center O

Step 2: Draw a diameter AB (horizontal line through center)

Step 3: Construct a perpendicular bisector of AB through center O

Step 4: Mark points C and D where the perpendicular intersects the circle (vertical diameter CD)

Step 5: Connect the four points in order: A to C, C to B, B to D, D to A

✓ ACBD is a square inscribed in the circle

Note: The diameters AB and CD are the diagonals of the square

What is an Incircle?

The incircle (inscribed circle) is the largest circle that fits inside the triangle and touches all three sides.

The center is called the INCENTER (intersection of angle bisectors)

Construction Steps

Step 1: Draw triangle ABC with any dimensions

Step 2: Construct the angle bisector of angle A:

• Place compass at A, draw arc intersecting sides AB and AC

• From these intersections, draw equal arcs inside the triangle that intersect

• Draw line from A through this intersection

Step 3: Construct the angle bisector of angle B (using same method)

Step 4: Mark point I where the two angle bisectors intersect (this is the INCENTER)

Step 5: From point I, construct a perpendicular to any side of the triangle

Step 6: Mark point P where the perpendicular meets the side

Step 7: Using I as center and IP as radius, draw the incircle

✓ This circle is tangent to all three sides of the triangle

Formula for Inradius

r = A / s

A = Area of the triangle

s = Semi-perimeter = (a + b + c) / 2

What is a Circumcircle?

The circumcircle (circumscribed circle) is the circle that passes through all three vertices of the triangle.

The center is called the CIRCUMCENTER (intersection of perpendicular bisectors)

Construction Steps

Step 1: Draw triangle ABC with any dimensions

Step 2: Construct the perpendicular bisector of side AB:

• Place compass at A, draw arc above and below AB

• With same radius, place compass at B and draw arcs intersecting previous arcs

• Draw line through the two intersection points

Step 3: Construct the perpendicular bisector of side BC (using same method)

Step 4: Mark point O where the two perpendicular bisectors intersect (this is the CIRCUMCENTER)

Step 5: Measure the distance from O to any vertex (OA, OB, or OC — they're all equal)

Step 6: Using O as center and this distance as radius, draw the circumcircle

✓ This circle passes through all three vertices A, B, and C

Formula for Circumradius

R = (abc) / (4A)

a, b, c = Lengths of the three sides

A = Area of the triangle

💡 Key Properties

• Incenter: Always inside the triangle; equidistant from all three sides

• Circumcenter: Can be inside, on, or outside the triangle; equidistant from all three vertices