Area of a triangle from its vertices, and why three collinear points give zero

0:00Three corners. One triangle. How much is inside it? There is an answer you have already met, and here it is, whole. A half of a bracket holding three pieces. Each piece is one corner's first coordinate, multiplied by the difference of the other two corners' second coordinates. Read it once and it looks fine. Try to write it out from memory tomorrow. Which corner goes with which difference? Which way round is each subtraction? Where exactly does the minus sit?

0:34Nothing about it is wrong. It is simply shapeless, and shapeless things do not survive being carried around in a head. So we are going to rewrite it. Not replace it. Rewrite it. And the rewrite will turn out to do a second job that the original could not do at all. Take the three corners and stack them, one to a row. First corner on the top row, its two coordinates side by side. Second corner underneath it. Third corner underneath that.

1:09Three rows of two numbers. An array wants to be square, and this one is a column short. Fill that column with ones. Every row now ends in a one. Put vertical bars round the whole thing, take its value, halve it, and there is your area. Three rows, one for each corner. Two coordinates in the order you always write them. And a one at the end of every row.

1:36That, you can carry. It is a picture rather than a pattern of subscripts. And it is the same arithmetic. Not similar. The same. Written out in letters, taking that array apart down the column of ones gives six terms, and those six terms are the old expression, term for term and sign for sign. So why ones? Almost everyone assumes they are padding, dropped in to make the array square.

2:07They are not padding. They are doing arithmetic. Watch what happens when you take the array apart down that third column. Each one multiplies a two by two, and every one of those two by twos is built out of the OTHER two corners. Each of them produces a difference. That is where the differences in the old expression came from. The ones are what turn coordinates into differences. Change the filling and watch it break.

2:38Fill the third column with twos instead. The whole thing doubles. On every one of fifteen thousand six hundred and twenty-five triangles checked, exactly double. So a column of twos is not a rival formula. It is this formula with the half thrown away, wearing a different coat. Now try something worse. Make the third column a copy of the first, so each corner's first coordinate appears twice. Multiply that out in letters and there is nothing left at all. Zero terms. Every triangle in the world comes out with area nothing.

3:20The ones are not padding. Move off them and the thing stops being an area. Here is the trouble with a formula that works. Checking it against itself proves nothing whatsoever. So we will not check this one against itself. We will run seven rules side by side and ask all seven the same questions. One: the rule we just built. Half the size of the value. Two: the same thing without the half.

3:51Three: the same thing without the bars, so the value keeps its sign. Four: the ones moved to the front column instead of the back. Five: the column filled with twos. Six: the third column copying the first. Seven: the old expression mis-copied, with one of its three signs turned over. That is not a silly rule. That is the mistake the shapelessness invites. And four questions, kept strictly apart.

4:21Does it give the area? Is it never negative? Is its value nothing exactly when the three corners lie on one straight line? And do all six ways of writing down the same three corners give one answer? Seven hundred and twenty-nine triangles, built from a small square of corner positions. Every rule, every question, every triangle. The rule we built passes all four, seven hundred and twenty-nine out of seven hundred and twenty-nine, four times over.

4:58So does the reading with the ones moved to the front. That one is not wrong. On all fifteen thousand six hundred and twenty-five triangles of the wider sweep, moving the ones to the front changes the answer zero times. Two of the seven get the area right. Two. Dropping the half gets it right on two hundred and seventy-three, and filling the column with twos gets it right on the same two hundred and seventy-three. The two are not merely alike. Zero triangles tell them apart.

5:35The two hundred and seventy-three are the flat ones, where doubling nothing is still nothing. Now look at the third column of the table, and something odd happens. Five of the seven pass the flatness test perfectly. Five. Dropping the bars passes it on all seven hundred and twenty-nine. Dropping the half passes it. The twos pass it. That matters more than it looks. It means passing the flatness test is cheap, and a rule can detect a straight line perfectly while handing you the wrong area every time.

6:13The unbarred reading is the sharpest case. Right on five hundred and one. Never negative on the same five hundred and one. Perfect on flatness. And steady under reordering on only two hundred and seventy-three. The copied column is the opposite. Never negative on all seven hundred and twenty-nine, steady on all seven hundred and twenty-nine, and it detects nothing, because it says every triangle is flat. And the mis-copy fails everything by a little: right on four hundred and ninety-seven, flatness on six hundred and twenty-nine, steady on three hundred and ninety-three.

7:01Take one triangle and write its three corners down in all six possible orders. Six arrays. Six values. The size never moves. On all seven hundred and twenty-nine triples, the six orderings give one single size. The sign does move. For a triangle with real area, exactly three of the six orderings come out positive and three come out negative. Four hundred and fifty-six of the triples split three and three. The remaining two hundred and seventy-three give nought six times over, because they are flat and nothing has a sign.

7:43So the sign is not telling you anything about the triangle. It is telling you the order in which you happened to write the corners down. Which is why the bars are there. An area is a quantity of surface. It cannot be negative. The value can be, and routinely is. Say that carefully, because it is the sentence students garble. It is the VALUE that may come out negative. It is the AREA that may not.

8:13A negative value is not a mistake. It is a reading of the order you chose. And you do not need to choose any particular order. There is no anticlockwise rule to remember here. Write the corners in any order at all, and the bars clean up after you. One more time: how do we know the formula is right, rather than merely self-consistent? Two other routes to the same area, neither of which ever builds an array.

8:45The first is a distance route. Take the three side lengths, square them, and feed the three squares into a relation that returns sixteen times the square of the area. It never sees a coordinate on its own. It never sees an order. It cannot produce a sign, and no square root is ever taken, so nothing is rounded. The second is a slicing route. Sort the three corners left to right. The long edge spans the whole width; at the middle corner's position, read the height of that edge.

9:22The vertical gap between that height and the middle corner cuts the triangle in two. Each piece is half a base times a height. Add them. No array. No product of coordinates from different corners. Just a sort, an interpolation and two schoolroom triangles. Those two routes agree with each other on all fifteen thousand six hundred and twenty-five triangles, with zero disagreements. And our array rule matches both of them at once on all fifteen thousand six hundred and twenty-five, and matches neither or only one on zero of them.

10:03That is the claim. Not that the formula agrees with itself. Now for the trap that the bars set. So far the corners were given and the area was wanted, and the bars threw the sign away with no loss at all. Turn the problem round. The area is handed to you, and one coordinate of one corner is missing. Now you must NOT throw the sign away, because you do not yet know which sign the value had.

10:34Half the size of the value equals the given area. Which means the value itself equals plus twice the area, or minus twice the area. Both are live. Both give a legitimate triangle. They are mirror images across the line joining the other two corners, and they enclose the same amount of surface. This is where the marks go. Take the size too early, keep one sign, and you lose exactly one of the two answers, every single time.

11:08It is the same instinct that served you well a minute ago, applied one step too soon. And now the second job, the one the old expression could not do. Slide the third corner towards the line through the other two. The triangle gets thinner. The area shrinks. The moment the third corner arrives on that line, the triangle has no inside left. Its area is nothing. So the value is nothing.

11:39Read that backwards and you have a test. Three corners lie on one straight line exactly when their array comes out nought. That single sentence is why this belongs with determinants at all, rather than with shapes. A question about geometry — do these three points line up? — has become a question of arithmetic. Work out one number and look at it. Of the seven hundred and twenty-nine triples, two hundred and seventy-three come out nought.

12:12And be careful with what that means. Nought does not mean the three points are the same point. Forty-eight of those two hundred and seventy-three are three genuinely different points, strung out along a line at different places on it. Flat includes coincident. It does not require it. One triangle, worked all the way through. Corners at three and eight, at minus four and two, and at five and one.

12:43Three rows. Three and eight and one. Minus four and two and one. Five and one and one. Take it apart along the top row. Three multiplies the two by two left when its row and column go: that comes to one. Three times one is three. Eight multiplies the next one, which comes to minus nine, and it arrives with a minus in front of it. Minus eight times minus nine is plus seventy-two.

13:14And the one at the end of the top row multiplies the last two by two, which comes to minus fourteen. Three, plus seventy-two, minus fourteen. Sixty-one. Halve it. The area is sixty-one by two. Leave it as a fraction. Do not reach for a decimal — the exact value is shorter and truer than anything you would round it to. Both independent routes were run on this same triangle. The slicing route returns sixty-one by two. The distance route returns fourteen thousand eight hundred and eighty-four, which is sixteen times the square of sixty-one by two.

13:59Now the trick, and it is a genuinely good one. You want the equation of the straight line through two given points. Take those two points, and add a third row holding a general point — call its coordinates x and y — a point that is allowed to be anywhere. Now declare the area to be nothing. That is not an assumption pulled out of the air. It says exactly this: the general point is on the line through the other two. Flat triangle. Value nought.

14:33Work the array out with the two named points at one and three, and at the origin. The value comes to three x minus y. Set it to nothing and you have y equals three x. Stop and notice what just happened. Nothing about lines was put in. No gradient. No intercept. No standard form. An area was declared to be nothing, and the equation of a line fell out of the arithmetic.

15:03Check it if you like. Between minus three and nine the whole-number points satisfying it are minus one and minus three, the origin, one and three, two and six, and three and nine. Every one of them lies on that line. Same two points. Now a different question. Put the third corner somewhere on the flat axis, at some first coordinate and second coordinate nothing. Slide it along until the triangle has area three. Where does it have to be?

15:38Build the array with the unknown in the bottom row and work it out. The value comes to three times the unknown. Half of that has to equal three. And here the last scene's warning arrives. Half the SIZE of three times the unknown is three, so three times the unknown is plus six or minus six. The unknown is plus two, or minus two. Two answers, and both of them are real triangles. One sits to the right of the origin, one to the left. They share a base and lean opposite ways.

16:16Same area. Same three sentences of working. Different triangles. Had you taken the size too early, you would have written down one of them and never known the other existed. The test survives letters, which is where it earns its keep. Three corners built out of three letters. First corner: a, and then b plus c. Second: b, and then c plus a. Third: c, and then a plus b.

16:46No numbers anywhere. Are they on a line? Build the array — three rows, ones down the third column — and multiply it out in letters, exactly as before. Every term cancels. Not most of them. All of them. Zero terms survive. So those three points lie on one straight line whatever a, b and c happen to be. There is no choice of three numbers that breaks it. And a control, so you know the multiplying-out was actually looking: nudge one of the six entries by one, leaving the shape otherwise untouched.

17:25Now it does not vanish. What is left is b minus a, which is nothing only for particular letters, not for all of them. That is the difference between an expression that is nought and an expression that is nought identically, and it is the whole content of a collinearity proof. Questions on this come in four shapes, and it is worth being able to recognise which one is in front of you.

17:54Shape one: corners given, area wanted. Straight through the formula. Three of them come out at fifteen by two, forty-seven by two, and fifteen. Notice the third. Its value is minus thirty, and its area is fifteen. The bars did their work, and nothing went wrong. Shape two: a collinearity, proved in letters. The one we just did. Shape three: the area given, a coordinate wanted. Both signs. Two answers.

18:27Two differently arranged versions of that question both land on the same pair, nought or eight. That looks like a copying slip and it is not — the two arrangements really do share an answer set. Shape four: the line through two named points, by declaring an area to be nothing. One pair gives the second coordinate as twice the first. Another gives the first coordinate as three times the second.

18:57And one more of shape three, wearing a multiple choice. The value works out to fifty minus ten times the unknown, the area is prescribed, and the unknown comes out as minus two or twelve. Two answers again. If the options offered you one of them on its own, that option is a trap built out of exactly the mistake this video keeps circling. So what does all of that leave you holding?

19:28One array. Three rows, one per corner. Ones down the last column. Bars round it, and a half in front. And three things that travel with it, none of which is inside the array. That the sign belongs to the order you wrote the corners in, so the bars come last and the area is never negative. That when the area is given instead of wanted, both signs stay alive and there are two answers.

19:57And that a value of nought means the three corners are on one straight line, which turns a geometric question into an arithmetic one and gives you the equation of a line for free. Store the array and drop those three, and you have a formula you can recite and cannot use. Every question that carries marks lives in one of the three. What this does not do is tell you anything about the array itself — why the ones work, why the value is what it is, what else it can measure.

20:33That is a larger story and it is not this one. This one was a rewrite that earned its keep twice: once by being memorable, and once by turning straightness into a number you can check.