Triangle and Circle Properties in Design

Triangle inequality. Any two sides of a triangle add to more than the third side. Picture two short sticks hinged to the ends of a long one: if together they are shorter than the long stick, they cannot reach each other. If they are exactly as long, they can only meet by lying flat along it, which makes a straight line, not a triangle. Sticks of 3 cm, 4 cm and 8 cm fail ($3+4=7$, less than 8). Sticks of 5 cm, 7 cm and 11 cm work ($5+7=12$, more than 11). A quick test: add the two shortest sides and check that the total is more than the longest side. If it is, the other two checks pass automatically.

Exterior angle. Extend one side of a triangle past a corner. The exterior angle this makes equals the sum of the two interior angles at the other two corners (the "opposite" interior angles). Why: if the interior angles are $a$, $b$ and $c$, then $a+b+c=180^{\circ }$. The exterior angle at the corner with $c$ sits on a straight line with $c$, so exterior $+c=180^{\circ }$. Both expressions equal $180^{\circ }$, so the exterior angle $=a+b$. Largest side, largest angle. In any triangle, the longest side is across from the largest angle and the shortest side is across from the smallest angle. A wider opening between two sides pushes their far ends further apart. The same goes for equals: two equal angles sit across from two equal sides, which is what makes an isosceles triangle.

Triangle inequality theorem | Perimeter, area, and volume | Geometry | Khan Academy

• Centre $O$: the point the same distance from every point on the circle. That distance is the radius.
• Chord: a segment joining two points on the circle. A chord through the centre is a diameter, twice the radius.
• Arc: a piece of the circle between two points. Two points $A$ and $B$ cut the circle into two arcs.
• Central angle: an angle with its vertex at the centre, such as $\angle AOB$.
• Inscribed angle: an angle with its vertex on the circle and its arms along two chords, such as $\angle APB$. It "stands on" the arc $AB$ that lies on the far side from $P$.

1. The central angle is twice the inscribed angle. When a central angle and an inscribed angle stand on the same arc, the central angle is twice as big: $\angle AOB=2\angle APB$. In the figure the arc $AB$ gives a $100^{\circ }$ central angle and a $50^{\circ }$ inscribed angle. 2. Inscribed angles on the same arc are equal. Every inscribed angle standing on arc $AB$ is half of the same central angle, so they are all equal: move $P$ anywhere along the other arc and $\angle APB$ does not change. 3. The angle in a semicircle is a right angle. If $AB$ is a diameter, the central angle $\angle AOB$ is a straight angle, $180^{\circ }$, so any inscribed angle standing on it is $180^{\circ }÷2=90^{\circ }$. 4. The perpendicular from the centre to a chord bisects the chord. Join $O$ to $A$ and $B$: both are radii, so triangle $OAB$ is isosceles. Dropping a perpendicular from $O$ to $AB$ at $M$ splits it into two matching right triangles, so $AM=MB$.

Angle Relationships in Circles and Triangles (Grade 9 Destreamed, MTH1W)

Property 4 also works in reverse: the line through the midpoint of a chord, perpendicular to the chord, always passes through the centre. That gives a way to find the centre of any circle, for example to drill a hole in the middle of a round tabletop or to place the hub of a wheel in a logo. Draw two chords that are not parallel, find the midpoint of each, and draw a line through each midpoint perpendicular to its chord. The two lines cross at the centre. Property 3 gives a second method: set the corner of a carpenter's square on the edge of the circle. The points where its two arms cross the edge are the ends of a diameter, because a right angle on a circle always stands on a diameter. Do this twice; the two diameters cross at the centre.