Properties of Figures

Introduction: Welcome to the World of Geometry!

Hello everyone! Let's start our study of "Properties of Figures" in Mathematics A.
This unit isn't just about calculation; it’s about testing your "ability to discover the rules hidden within a figure." Even shapes that look complicated at first glance can be solved easily like a puzzle once you know a few theorems (rules).
It might feel difficult at first, but don't worry. As you draw the figures, we’ll go through the key points one by one!

1. Properties of Triangles: Master the Magic of Ratios

There are many useful theorems regarding side lengths and ratios in triangles. Let's look at three particularly important ones.

① The Angle Bisector Theorem

When there is a line that splits one of the angles of a triangle perfectly in two, the following rule applies:
【Point】
In \(\triangle ABC\), if the bisector of \(\angle A\) intersects side \(BC\) at point \(D\):
\(AB : AC = BD : DC\)
Image: Remember it with the rhythm "Left side : Right side = Bottom left : Bottom right"! The same ratio holds for exterior angle bisectors, but let’s master this interior angle version first.

② Ceva’s Theorem and Menelaus’s Theorem

These two are frequent flyers on the "Common Test"!
Ceva’s Theorem: Used when there is a point inside a triangle and lines are drawn from each vertex through that point.
Menelaus’s Theorem: Used when a single line "skewers" a triangle.

【Study Tip: The Fox's Walk】
In both theorems, the form of the equation is \(\frac{x}{y} \cdot \frac{z}{w} \cdot \frac{u}{v} = 1\).
Just trace a path around the figure like you're taking a walk: Vertex → point on side → Vertex → point on side… Keep going until you make a full loop. Even if you see a shape that looks like a "fox face" (Menelaus), the rule remains the same!

③ The Five Centers of a Triangle (Centroid, Circumcenter, Incenter)

A triangle has five special "centers," but let's start by distinguishing these three:
・Centroid (G): The intersection of the medians (lines connecting a vertex to the midpoint of the opposite side). The ratio is always 2:1!
・Circumcenter (O): The center of the circumscribed circle. Its defining feature is that it is equidistant from all vertices.
・Incenter (I): The center of the inscribed circle. It is equidistant from all sides (the radius) and is the intersection of the angle bisectors.

【Summary: Key Points of Triangles】
・When you see an angle bisector, write down the "ratios"!
・For complex ratio problems, it's time for "Ceva and Menelaus"!
・Whenever you see a centroid, remember "2:1" immediately!

2. Properties of Circles: From Circumangles to the Power of a Point

In circle problems, changing your perspective is often all you need to see the answer.

① Cyclic Quadrilaterals

When a quadrilateral fits perfectly inside a circle, there are some amazing rules:
・The sum of opposite angles is \(180^\circ\)
・One exterior angle is equal to the opposite interior angle
Common Mistake: This doesn't apply to just any quadrilateral! It’s a "special move" that you can only use when all four vertices lie on the circle.

② The Alternate Segment Theorem

This theorem concerns the angle formed by a tangent to a circle and a chord through the point of contact.
"The angle between the tangent and the chord is equal to the angle in the alternate segment (the circumference angle)."
It sounds difficult in words, but looking at the diagram, it’s as if "the triangle's angle has hopped over to the tangent line outside."

③ The Power of a Point Theorem

A rule for multiplying lengths when two lines intersect inside (or outside) a circle.
If two lines drawn from point \(P\) intersect the circle at points \(A, B\) and \(C, D\), then:
\(PA \cdot PB = PC \cdot PD\)
Study Tip: Always remember to multiply "the distance from the intersection point \(P\) to the circle" twice. The formula is the same whether the intersection point is inside or outside the circle!

【Trivia】
"Power" here relates to the power of a point, but you can think of it as the area of rectangles being formed. It turns out this theorem comes from the property that the areas of the rectangles created by the two lines are equal.

【Summary: Key Points of Circles】
・When you see a circle, look for "equal angles (circumangles)" first!
・If a tangent appears, suspect the "Alternate Segment Theorem" or "perpendicularity to the radius."
・When there is a lot of information about lengths, it’s a chance to use the "Power of a Point Theorem"!

3. Construction and Spatial Figures: Imagine the 3D World

Finally, let's step out of the plane and talk about 3D shapes.

① Positional Relationships between Lines and Planes

If you aren't good at spatial geometry, try imagining everyday objects.
・How do you stand a pencil (a line) upright on a desk (a plane)?
・If a line \(l\) is perpendicular to two intersecting lines on a plane \(\alpha\), then line \(l\) is perpendicular to the entire plane \(\alpha\).
Example: It's the same principle as a tripod standing firmly on the ground!

② Regular Polyhedra

In this world, there are only 5 types of "regular polyhedra" where all faces are the same regular polygon and the same number of faces meet at every vertex.
Regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron, and regular icosahedron.
On the Common Test, you might be asked about the "shape of the faces" or the "number of vertices" for these.

③ Euler's Polyhedron Formula

For any convex polyhedron, the following formula holds:
(Number of vertices) - (Number of edges) + (Number of faces) = 2
Remember it as \(V - E + F = 2\) (V: Vertex, E: Edge, F: Face).
Mnemonic: "V (Victory) minus E (the) plus F (faces) equals 2."
This is the ultimate tool for checking your work if your calculations don't seem to add up.

【Summary: Key Points of Space】
・For spatial problems, extract only the necessary plane and think in "2D"!
・There are only 5 types of regular polyhedra. Remember their names and shapes as a set!
・If you're confused about the number of vertices, edges, or faces, use "Euler's Formula" to double-check!

Closing: How to Become Great at Geometry

Great work on completing this study of the properties of figures!
The secret to solving Math A geometry problems is to "keep writing down what you know on your diagram." Lengths, ratios, angles, marking equal segments… the more you move your pencil and make your diagram lively, the more the path to the answer will reveal itself.

You might take a while to recall the theorems at first, but with practice, you'll reach a point where you see a figure and immediately flash, "Oh, this is Menelaus!" Go at your own pace, one step at a time. I'm rooting for you!