Say the construction and get a draggable figure

Say the construction in ordinary English. GoodMaith builds it on a live board in the same chat. Drag a point, and the objects that depend on it update. The next sentence can add a line, a foot, or a second figure.

In GeoGebra the same figures are a sequence of tools. A circle through three points is a tool and three clicks. A triangle from two sides and the included angle is a segment, a rotation, and a polygon. You have to know which tool, and in which order. Here you say the relation you already mean, and the board builds the objects that relation forces.

A circle through three points

Name the points. On this board A is $(0,0)$, B is $(4,0)$, and C is $(1,3)$. Three points that are not on one line determine one circle. The circle is the object that has to pass through all three, so it is not a circle drawn by eye around them. Its center is the point equidistant from A, B, and C, which is where the perpendicular bisectors of the sides meet. You do not have to say that sentence for the circle to appear. The construction knows it.

Drag C straight up. A and B stay where you left them, and the circle grows so that it still goes through the new C as well as through A and B. Drag B along the horizontal axis, closer to A. The circle changes again, and C is still on it. Drag A off the axis. All three points remain on the circle, because that is how the circle was built. If the circle were a finished drawing, moving C would either leave a gap or drag a picture that no longer means "through these three." The same check is the one in Good math, made visible.

The next sentence can use the circle that is already there. "Mark the center" puts the equidistant point on the board. "Draw the perpendicular bisector of AB" adds the line that the center has to lie on. Those objects are new, and they depend on A, B, and C, so the next drag of C moves them too. The figure on this page stops at the circle and the three points, so you can see the relation before the extra lines arrive. In the chat they arrive on that same board.

Two sides and the included angle

A triangle can start from measures instead of from three placed points. Draw triangle ABC with AB = 5, AC = 7, and angle A equal to 60 degrees, a turn of $\pi /3$.

A is fixed at the origin. The base is the horizontal line through A. B starts on that line at $(5,0)$, so AB begins at $5$. C is not placed by guessing a height. A point P sits at $(7,0)$, seven units from A, and C is the image of P under a rotation by $\pi /3$ about A. That rotation keeps the distance, so AC is $7$. It turns the direction from A through one third of a straight angle, so angle A starts at $60$ degrees. C lands at $(7\cos \pi /3,7\sin \pi /3)$, which is $(3.5,7\sqrt{3}/2)$. The polygon is A, B, C. The angle mark on the board is that turn at A.

Drag B along the base. While B stays on the positive side of A, the angle at A does not change, and C does not move, because C was built from the rotation of P, not from B. AB is the length you are changing. Slide B out past $5$ and the side grows. Slide B in toward A and the side shrinks. The given angle and the given side AC are still the ones you asked for. Slide B through A onto the negative side and you are no longer looking at the angle you named. The turn from the positive base to AC is still $\pi /3$, but B is on the other ray, so the angle the board marks between B, A, and C is the other one. That is why the caption tells you to keep B on the positive side.

The next sentence fits this triangle the way the center fits the circle. "Mark the midpoint of BC" adds a point that has to stay halfway between B and C, so it follows B when you drag. "Drop a perpendicular from A to BC" adds the foot on the line BC. Neither of those is drawn yet on the figure above. Ask in the chat and they are built from A, B, and C rather than pasted on top.

You can try the tutor without an account. Sign in to keep the chats. The graphing workbench is the same kind of board when you already know the construction and want to drive it yourself, with no tutor choosing the next object. GoodMaith is an AI math solver with live graphs. A function sentence, with a curve and a shift, is the companion page describe a math problem and get an interactive graph. To start from a sentence of your own, chat with a live board.