What is dynamic geometry software?
Dynamic geometry software (for example GeoGebra or similar tools) lets you draw points, lines, circles and measure them on a screen. The special thing is that when you drag a free point, everything built from it moves too, and keeps its rule.
- Free object: you can drag it (like A, B and C).
- Dependent object: built from other objects (like the midpoint or the circle). It follows.
- Measure: length, angle, area. They update live.
Invariants: what never changes
An invariant is something that stays the same while you drag. In any triangle, the three angles always add to 180 degrees. Each angle changes, but the sum does not.
Method: 1) build the figure, 2) measure, 3) drag to many positions, 4) note what stays constant, 5) write a guess (a conjecture), 6) try to break it, 7) prove it.
Constructions: circumcentre and circumcircle
A perpendicular bisector of a segment is the line that cuts it in half at a right angle. Every point on it is the same distance from both ends.
Draw the perpendicular bisectors of AB, BC and CA. They meet at one point, the circumcentre O. Since O is on the bisectors of AB and BC, OA = OB = OC. So a circle centred at O passes through all three corners. It is the circumcircle, and its radius R = OA.
Geometric applications
- Find the centre of a circle from three points on it (like a broken plate).
- Right angle in a semicircle: if C is on the circle with diameter AB, the angle at C is always 90 degrees. In the scene, when the angle at C is 90 degrees, O is exactly the midpoint of the longest side.
- Location problems: the point equally far from three villages is a circumcentre.
- Locus: the path a point makes when another moves (the purple trail).
Testing a guess is not proving it
Dragging checks many cases, but not all. A guess that works in 100 positions might fail in a special case. A proof uses reasons that hold for every case. Use the software to find and test ideas, then use reasoning to be sure. Software also has rounding: it shows 180 when the true value is exactly 180, but it may show 179.99 in a different tool.
Try it: break the rule
In step 6 move C with both sliders. Try to find a triangle where the angle sum is not 180 degrees. You cannot. Next, try to find a triangle where O lies outside the triangle (hint: make the angle at C bigger than 90 degrees). Where is O when the angle at C is exactly 90?
Key formulas and definitions
- Angle sum of a triangle = 180 degrees
- Midpoint of (x1, y1) and (x2, y2) = ((x1 + x2)/2, (y1 + y2)/2)
- Circumcentre: OA = OB = OC = R
Worked examples
1. Two angles of a triangle on screen show 50 and 70 degrees. What does the third show?
Angle sum is an invariant: 180 - 50 - 70 = 60 degrees.
2. A right triangle has a hypotenuse of 10. Where is its circumcentre and what is R?
O is at the midpoint of the hypotenuse, so R = 10/2 = 5.
3. Find the midpoint of A(2, 4) and B(8, 10).
((2 + 8)/2, (4 + 10)/2) = (5, 7).
Common mistakes
- Thinking that a few dragged positions prove a rule. They only test it.
- Dragging a dependent point. Only free points can be dragged; dependent ones follow.
- Assuming the circumcentre is always inside the triangle. For an obtuse triangle it is outside.
- Mixing up the midpoint with the perpendicular bisector. The bisector is a whole line, the midpoint is one point on it.